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Lowest Common Multiple

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Jeremy Bansil
Yash Singhal
Mateo Matijasevick
Chung Kevin

The lowest common multiple (LCM) of a finite set of non-zero integers is the smallest positive number that is a multiple of each integer in the set. It is a fundamental concept in number theory, and is closely related to the greatest common divisor . It is taught to elementary school students as an aid in adding fractions: to simplify $ \frac{a}{x}+\frac{b}{y}, $ one first writes the fractions with a common denominator, which is the LCM of $x$ and $y.$

Simplify $ \frac1{105} + \frac1{50}.$ Since $ \text{lcm}(105,50) = 1050,$ this sum can be rewritten as $ \frac{10}{1050} + \frac{21}{1050} = \frac{31}{1050}.$ $_\square$

Find the lcm of $ 30$ and $36.$ Here is a list of the positive multiples of each number: \[ \begin{align} 30: &30,\ 60,\ 90,\ 120,\ 150,\ 180, \ldots \\\\ 36: &36,\ 72,\ 108,\ 144,\ 180, \ldots. \end{align} \] The first number that appears on both lists is $ 180.$ $_\square$

The LCM is important in many applications of elementary number theory, including the Chinese remainder theorem , where $ \text{lcm}(a,b)$ appears as the modulus of the unique solution to a system of equations modulo $a$ and $b.$

Computing the LCM

Connection with the gcd, properties of the lcm, problem solving.

The LCM in the introduction was computed by listing the multiples of each number and searching for the first integer on every list. (The LCM always exists because the product of a list of numbers is divisible by each of the numbers in the list.) This is a very ineffective way of computing the LCM in general.

However, if the prime factorization of the numbers is known, then computing the least common multiple is much simpler. The primes in the factorization of the LCM are the primes that appear in the factorizations of at least one member of the list, and their exponent is the maximum of the exponents that appear in the individual factorizations.

Compute $\text{lcm}(4200,3780,3528)$ using prime factorizations. We have \[ \begin{align} 4200 &= 2^3 \cdot 3 \cdot 5^2 \cdot 7 \\ 3780 &= 2^2 \cdot 3^3 \cdot 5 \cdot 7 \\ 3528 &= 2^3 \cdot 3^2 \phantom{\cdot 5 \cdot} \cdot 7^2. \end{align} \] The LCM can be read off from these factorizations by taking the maximum exponent for each prime: $ 2^3 \cdot 3^3 \cdot 5^2 \cdot 7^2 = 264600.$ $_\square$

Generalizing this example, if the prime factorizations of $a$ and $b$ are

\[\begin{align} a & = p_1 ^{\alpha_1} p_2 ^{\alpha_2} \ldots p_k ^{\alpha_k} \\\\ b & = p_1 ^{\beta_1} p_2 ^ {\beta_2} \ldots p_k ^ {\beta_k}, \\ \end{align} \]

where the $ p_i$ are distinct primes and the $\alpha_i$ and $ \beta_i$ are nonnegative integers, then

\[ \mbox{lcm}(a,b) = p_1 ^{\max(\alpha_1, \beta_1)} p_2 ^{\max(\alpha_2, \beta_2)} \ldots p_k ^{\max(\alpha_k, \beta_k)}. \]

A similar formula holds for finding the LCM of several integers, by taking the largest exponent for each prime.

What is the smallest positive integer $n$ such that $\mbox{lcm}(n, 30) = 180?$ The prime factorization of $30$ is $30 = 2 \times 3\times 5 $ and the prime factorization of $180$ is $180=2^2\times 3^2\times 5$. Then $n$ must include a factor of $2^2$ and a factor of $3^2$, so the smallest positive integer $n$ is $n=2^2 \times 3^2 = 4 \times 9 = 36$. $_\square$

Recall that the greatest common divisor of a set of integers is the largest (positive) number which is a divisor of each integer in the set. The two concepts are intimately related; in particular, they satisfy the following theorem:

\[ \gcd(a,b) \times \mbox{lcm}(a,b) = ab \]

This is straightforward from the descriptions using prime factorizations, but it can also be proved directly. Consider the integer $m = \frac{ab}{\gcd(a,b)};$ since it equals $\frac{a}{\gcd(a,b)} b,$ and the first fraction is an integer, it is a multiple of $b;$ similarly, it is a multiple of $a$ since it can be rewritten as $a \frac{b}{\gcd(a,b)}.$ So $m$ is a common multiple of $a,b.$ This shows that $\text{lcm}(a,b) \le m,$ and multiplying through by the denominator of $m$ gives $\gcd(a,b) \times \text{lcm}(a,b) \le ab.$ But a similar "dual" argument shows the reverse inequality: let $d = \frac{ab}{\text{lcm}(a,b)}.$ Then $d \left( \frac{\text{lcm}(a,b)}a \right) = b,$ and the quantity in parentheses is an integer, so $d|b.$ Similarly $d|a.$ So $ d\le \gcd(a,b),$ and multiplying through by the denominator of $d$ gives $ ab \le \gcd(a,b) \times \text{lcm}(a,b).$ These two statements together show that the two sides are equal. $_\square$

Given that $a$ and $b$ are 2 integers such that $ 13 \gcd(a,b) = \mbox{lcm}(a,b) $ and $ a + b = 2016$, what are the values of $a$ and $b?$ Let $ G = \gcd(a,b)$ and $ L = \mbox{lcm}(a,b)$. Let $ a = a^* G$ and $ b = b^* G,$ where $ \gcd(a^*, b^*) = 1 $ by construction. Since $ (a^*G)\times (b^*G) = ab = GL = G \times 13 G ,$ we get that $ a^* b^* = 13. $ Hence, we have $ \{ a^*, b^* \} = \{ 1, 13\} .$ Without loss of generality, we may assume that $a\leq b,$ and thus $ a = G, b = 13 G $. Since $ 2016 = a + b = G + 13G = 14 G, $ $G = \frac{2016} { 14} = 144 $. Thus, $ \{a, b\} = \{ 144, 1872 \} $. $_\square$

This relationship with the gcd also gives an efficient algorithm for computing the LCM that does not require prime factorization. Since the Euclidean algorithm computes $ \gcd(a,b)$ quickly, the value of $ \text{lcm}(a,b)$ can be recovered as $ \frac{ab}{\gcd(a,b)}.$

The properties of the LCM are dual to those of the GCD, and many of them can be deduced from the explicit relationship between the two proved above.

The LCM of a list of numbers divides any other common multiple.

This is clear from the formula via prime factorization; it can also be deduced from the analogous property of the GCD.

The intersection of the sets of multiples of the individual numbers in a list equals the set of multiples of their LCM.

This is essentially a statement about ideals in the ring of integers.

The LCM of a list of numbers can be computed two at a time, e.g. $ \text{lcm}(a,b,c) = \text{lcm}\big(\text{lcm}(a,b),c\big).$

So algorithms that compute the LCM can concentrate on the LCM of a pair of numbers since the LCM of a larger list can be computed from LCMs of pairs.

The traffic lights at three different intersections change after every 48 seconds, 72 seconds, and 108 seconds, respectively. If they change simultaneously at 9 a.m., then when is the next time that they change simultaneously?

For the end-of-year examinations, Stella wants to bring a bag of candies to share with every student who earns an "A" in her class.

She knows that if there are 2, 3, 4, 5, 6, or even 7 top students, then she should be able to distribute the candies out evenly, and she doesn't want to take any candies home with her.

Given that she is going to bring at least 10 candies, what is the minimum number that she must bring?

Cicadas live underground for most of their lives and only emerge in the spring of their last year in order to mate and reproduce. The North American Genus of Cicada, known as Magicicada, has an extremely long life cycle of 13 or 17 years. Each brood of cicadas has its own emergent years and life cycles.

The state of Kansas only gets 2 broods of Magicicada. The first brood last appeared in 1998 and has a life cycle of 17 years. The second brood last appeared in 2011 and has a life cycle of 13 years. (It is now 2014.) How many years will it take for both broods to emerge together?

Image credit: Instagram user brandycandy, Data: Magicicada.org

The 6-digit number $\overline{739ABC}$ is divisible by 7, 8, and 9. What values can A, B, and C take?

Enter your answer as the sum of the sums $A+B+C$ for all triples $(A,B,C)$ that work.

Find the smallest positive integer which when divided by

9 leaves remainder 8,
8 leaves remainder 7,
7 leaves remainder 6,
6 leaves remainder 5,
5 leaves remainder 4,
4 leaves remainder 3,
3 leaves remainder 2, and
2 leaves remainder 1.

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Least Common Multiple (LCM)

The fascinating realm of mathematics with our comprehensive guide on the Least Common Multiple (LCM). This essential concept, pivotal for seamless calculations in arithmetic, algebra, and beyond, serves as the cornerstone for solving diverse problems involving fractions, equations, and number theory. Unravel the mysteries of LCM through illustrative examples, unlocking efficient problem-solving techniques. Embrace this journey to elevate your mathematical proficiency and navigate through numerical challenges with ease.

What is the Least Common Multiple (LCM)?

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. It’s the minimum shared value that all given numbers divide into without leaving a remainder.

Consider the numbers 4 and 5. To find the LCM of 4 and 5, we list their multiples:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …

Looking at these lists, the smallest number that appears in both lists is 20. Therefore, 20 is the Least Common Multiple of 4 and 5. This means 20 is the smallest number that both 4 and 5 can divide into without leaving any remainder.

How to Find Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all these numbers. It’s a fundamental concept in mathematics, particularly useful in solving problems involving fractions, algebra, and number theory. Understanding how to find the LCM can simplify calculations and help in comparing or combining different quantities.

Before diving into methods for finding the LCM, it’s essential to grasp its significance. The LCM is used to find a common denominator in fractions, which simplifies adding, subtracting, and comparing them. It’s also crucial in solving equations with multiple terms and in real-life situations like scheduling events that have different repeating cycles.

They are three ways to find Least Common Multiple(LCM) :

Prime Factorization Method
Listing Multiples Method
Division Method

1. Prime Factorization Method

Break Down Each Number : Decompose each number into its prime factors.
Identify Common and Unique Factors : For each number, list out all prime factors, considering the highest power of each prime that appears in any of the numbers.
Multiply the Highest Powers of All Prime Factors : The product of these factors will give you the LCM.

least common multiple by prime factorization tree method

To find the LCM of 12 and 18 using prime factorization:

Prime factors of 12: 2×2×3
Prime factors of 18: 2×3×3
LCM: 2²×3² = 36

2. Listing Multiples Method

List Multiples of Each Number : Start listing multiples of each number until you find the first common multiple.
Identify the Least Common Multiple : The first common multiple you encounter is the LCM.

For numbers 4 and 5, the multiples are:

4: 4, 8, 12, 16, 20, 24…
5: 5, 10, 15, 20, 25…

The LCM is 20, the first common multiple.

3. Division Method

Divide by Common Prime Numbers : Begin with the smallest prime number (usually 2) and divide the given numbers if possible. Write the quotient below each number.
Repeat Until No More Division is Possible : Continue dividing by prime numbers (2, 3, 5, etc.) until no numbers can be evenly divided.
Multiply All Divisors and Remaining Numbers : The product of all the divisors and the remaining numbers at the bottom is the LCM.

Finding the LCM of 14 and 20:

Divide by 2: Quotients are 7 (for 14) and 10 (for 20).
Divide by 2 again: Quotients are 7 and 5.
Multiply all divisors and remaining numbers: 2×2×7×5=140.

LCM Formula

The Least Common Multiple (LCM) of two numbers, a and b , can be found using the formula:

LCM(a, b)=∣a× b∣÷ GCD (a, b)

Here, ∣a× b∣ represents the absolute value of the product of a and b , and GCD(a, b) is the Greatest Common Divisor of a and b . This formula efficiently combines multiplication and division, leveraging the relationship between the LCM and GCD to find the smallest multiple common to both numbers.

let’s solve for the LCM of 15 and 20:

First, find the Greatest Common Divisor (GCD) of 15 and 20. The GCD of 15 and 20 is 5.
Then, multiply the numbers together: 15×20=300
Finally, divide the product by the GCD: 300 ÷ 5= 60

Therefore, the Least Common Multiple (LCM) of 15 and 20 is 60.

Properties of Least Common Multiple (LCM)

The Least Common Multiple (LCM) is a fundamental concept in mathematics that plays a pivotal role in various computational processes, including solving equations, simplifying fractions, and analyzing number sets. Understanding the properties of LCM not only enhances one’s mathematical prowess but also lays the groundwork for advanced mathematical applications.

Least Common Multiple (LCM) of Two Numbers

The Least Common Multiple (LCM) of two numbers is a foundational concept in mathematics that finds extensive application across various domains, from solving fraction-based problems to scheduling and planning. It represents the smallest number that is a multiple of both numbers, facilitating arithmetic operations and problem-solving with ease and efficiency. Grasping the LCM concept is crucial for students, educators, and professionals who deal with mathematical computations regularly.

To compute the LCM of two numbers, several methods can be employed, each suited to different situations and complexity levels. The most common approaches include the prime factorization method, where numbers are broken down into their prime components, and the division method, which systematically reduces numbers using common divisors. Choosing the right method depends on the numbers at hand and the solver’s comfort with mathematical operations.

To elucidate the concept, let’s consider an example of finding the LCM of 12 and 15 using the prime factorization method:

The prime factors are 2, 3, and 5.
The highest powers are 2² (from 12) and 3 and 5 (from 15).
LCM = 2² ×3×5=4×3×5=60

Therefore, the LCM of 12 and 15 is 60, meaning 60 is the smallest number into which both 12 and 15 can evenly divide.

Least Common Multiple (LCM) of Three Numbers

The Least Common Multiple (LCM) extends beyond pairs of numbers, applying to sets of three or more, serving as a crucial tool in mathematics for simplifying complex problems. The LCM of three numbers is the smallest number that all three numbers divide into evenly, facilitating operations involving fractions, scheduling, and more. Understanding how to compute the LCM for three numbers is vital for anyone looking to deepen their mathematical skills and apply them in various contexts.

Calculating the LCM of three numbers can be approached in multiple ways, with the prime factorization and division methods being the most common. The prime factorization involves breaking each number into its prime components and combining them to find the LCM, while the division method systematically reduces the numbers using common divisors. The choice of method often depends on the specific numbers and the solver’s preference for mathematical techniques.

Let’s illustrate the process with an example, calculating the LCM of 8, 9, and 21 using the prime factorization method:

The prime factors are 2, 3, and 7.
The highest powers are 2³ (from 8), 3² (from 9), and 7 (from 21).
LCM = 2³ ×3²×7=8×9×7=504

Thus, the LCM of 8, 9, and 21 is 504, the smallest number that all three integers divide into without leaving a remainder.

Relationship Between LCM and HCF

LCM and HCF are two sides of the same coin in mathematics, each serving a unique purpose but also complementing each other. While LCM determines the smallest common multiple shared by two or more numbers, HCF finds the largest factor that divides those numbers without a remainder. The relationship between them is not just theoretical but practical, aiding in problem-solving and mathematical reasoning.

To clearly understand the relationship between LCM and HCF, let’s explore their connection through a table that outlines the key aspects of their interrelation:

This table encapsulates the core aspects of LCM and HCF, illustrating their distinct roles and the mathematical formula that binds them together. It’s evident that while they serve different purposes, the relationship LCM ( a , b ) × HCF ( a , b ) = ∣ a × b ∣ showcases a direct and profound connection, emphasizing the balance and symmetry inherent in mathematics.

Least Common Multiple Chart

1. Find LCM of 6 and 8?

Combine 2³ (from 8) and 3 (from 6).
LCM = 8×3=24

The LCM of 6 and 8 is 24.

2. Find LCM of 14 and 20

Combine 4(from 20), 5 (from 20), and 7 (from 14).
LCM = =4×5×7=140

The LCM of 14 and 20 is 140.

3. Find LCM of 18, 24, and 30

Combine 2³ (from 24), 3² (from 18), and 5 (from 30).
LCM = 8×9×5=360

The LCM of 18, 24, and 30 is 360.

In conclusion, understanding the Least Common Multiple (LCM) is essential for simplifying mathematical problems and finding common denominators. Through methods like prime factorization, calculating the LCM becomes straightforward, aiding in diverse applications from fraction operations to scheduling tasks. Mastering LCM calculations not only enhances mathematical proficiency but also equips learners with critical problem-solving skills.

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Least Common Multiple Simple How-To w/ 9+ Examples!

// Last Updated: November 6, 2020 - Watch Video //

Did you know that the least common multiple, or LCM, is the smallest positive multiple that is common to two or more numbers?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching least common multiple

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

But first, what is a multiple, and how is it different than a factor?

Let’s look at an example.

List the multiples and factors of 6.

Multiples of 6 : 6, 12, 18, 24, 30, 36, 42, …
Factors of 6 : 1, 2, 3, and 6

So, factors are the integers we multiply together to get the original number.

And multiples are all the integers that 6 can go into.

That means, whenever you see the word “multiple” you want to think multiply!

How do we find the least common multiple?

Well, just like we saw with finding the greatest common factor , there are actually two techniques for identifying the LCM:

Prime Factorization

Using our listing method , we will work L-C-M backward:

M : List multiples for each number
C : Identify common multiples
L : Choose the lowest multiple (smallest number they have in common)

And for the prime factorization method , we will create a factor tree and identify each number’s prime factors. Then we will select all factors, always choosing the largest common factor.

List Method

The listing method, sometimes called the roster method, is simply a listing of multiples. When we learned to multiply we employed a technique called skip counting to help us recognize equal groups, and that’s basically what we are doing here — we’re going to skip count by each integer given.

For example, let’s find the least common multiple of 6 and 10 using both techniques.

First, we will use the listing method by “skip counting” and select the lowest integer they both have in common.

Find The LCM of the Numbers Using Lists of Multiples

Prime Factorization Method

Now, let’s see how to use the prime factorization method. For this technique, we create factor trees for each integer as seen below, and then multiply the largest of all prime values found in both trees.

Prime Factorization of 2 Numbers

Method Summary

Here’s a breakdown of the two methods for identifying the Least Common Multiple:

When we list factors, we choose the LOWEST common number.
When we use prime factorization, we choose MORE – most power from each factor.

As you have probably noticed, the first method is easier to use, as all we’re doing is multiplying (i.e., skip-counting) and selecting the lowest number.

But this method is not very practical for integers bigger than 12, whereas the prime factorization method is tried and true and will work for any integer as noted on the Art of Problem Solving .

Example — Hard LCM Problem

Let’s look at another example, where using prime factorization really shows itself to be the best method for discovering the least common multiple

Find the LCM of 8 and 42.

LCM of 42 and 8 — Example

LCM Using Prime Factorization

As you can see, prime factorization helped us to locate the LCM much faster than skip-counting.

Example — LCM Of Three Numbers

What about finding the LCM for three numbers?

Like we saw with the example above, the method we should employ when discovering the least common multiple for three numbers is prime factorization.

For example, let’s find the LCM of 90, 120, and 300

How To Find LCM of 3 Numbers Using Prime Factorization

Trying to list multiples for these three numbers would have been incredibly tedious. Aren’t you glad we just used prime factorization!

Worksheet (PDF) — Hands on Practice

It’s all fun and games until you try it on your own. So give these worksheets a go — and improve your knowledge!

Least Common Multiple — Practice Problems Least Common Multiple — Step-by-Step Solutions

Video Tutorial — Full Lesson w/ Detailed Examples

Together we will look at various examples of finding the least common multiple and ensuring that we can employ both methods (listing and prime factorization) with success.

Introduction to Video: Least Common Multiple
00:00:29 – How do you find the LCM?
00:08:02 – Determine the LCM using both methods (Examples #1-2)
00:13:02 – Find the least common multiple for two or three numbers (Examples #3-8)
00:30:03 – Uncover the LCM for three numbers using prime factorization (Examples #9-10)
Practice Problems with Step-by-Step Solutions
Chapter Tests with Video Solutions

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Least Common Multiple (LCM) Worksheets

This extensive collection of printable worksheets on LCM is designed and recommended for students of grade 5 through grade 8. The exercises covered in this module include finding common multiples, finding the least common multiple for a set of numbers and much more! Explore some of these handouts for free!

Common Multiples

List the first ten multiples for the given pair of numbers. Then, identify two common multiples for both numbers. Solve the LCM worksheets for 5th grade and 6th grade kids that contain five problems each.

Download the set

LCM: Descriptive

Learners are required to list the first ten multiples for each set of two numbers featured in the questions. Next, they need to determine the LCM by finding the smallest common multiple that is divisible by both numbers.

Finding LCM | Standard

Turn your young learners in grade 6 and grade 7 into math whizzes with this array of LCM worksheets involving numbers up to 25. Instruct them to find the LCM for each number pair.

Finding LCM | Standard - More Practice

Are you struggling with the concept of LCM? Fret not! You can breeze through this topic by finding the least common multiple of a set of two-digit numbers with our meticulously drafted worksheets.

GCF, LCM and the Product

Gain deep insight as you dive into this collection of GCF, LCM and Product worksheets, and rise up as stronger and wiser mathematicians. Our 3-part practice resources help strengthen your skill and perfect your technique!

LCM of Three Numbers

Find the least common multiple for each set of three numbers. Use the answer key to validate your responses. Each pdf worksheet has 10 sets of numbers for 7th grade and 8th grade students.

GCF and LCM Word Problems

Enjoy an abundance of GCF and LCM real-world scenarios that test your problem-solving skills and broaden your learning. Read the problems, analyze the data, and apply your skills to find the GCD and HCF correctly.

LCM of Polynomials

Figure out the least common multiple with these LCM of polynomials worksheets. Learn to find the LCM of two, three monomials and polynomials segregated into easy and moderate levels. Find the other polynomial in the given linear, quadratic or polynomial expression as well.

(35 Worksheets)

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HCF and LCM Questions

Here, students will find HCF and LCM questions with in-depth explanations that will aid in their understanding of the concept. In mathematics, LCM represents the Least Common Multiple and HCF represents Highest Common Factor. The HCF and LCM questions that we have provided here will assist students comprehend how to find the LCM and HCF of the given numbers. Different questions are also provided so that students can practise them, and they can compare their answers with the provided solutions on our page. For additional information on the HCF and LCM, click here .

HCF and LCM Questions with Solutions

1. Find the HCF of 24 and 36.

To find HCF of 24 and 36, first list the factors of 24 and 36.

Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12 , 18, and 36.

Here, the greatest factor that is common in both the factors list is 12.

Thus, 12 is the greatest number that divides both 24 and 36 evenly.

Therefore, the HCF of 24 and 36 is 12.

2. Determine the GCF of 18 and 48.

To find: HCF (18, 48)

The factors of 18 are 1, 2, 3, 6 , 9 and 18

The factors of 48 are 1, 2, 3, 4, 6 , 8, 12, 16, 24 and 48.

Thus, 6 is the greatest number that divides 18 and 48 completely.

Hence, the GCF/HCF of 18 and 48 is 6.

3. Find the HCF of 135 and 225 using the prime factorisation method.

To find the HCF of 135 and 225 using the prime factorisation method, follow the below steps:

Thus, the prime factorisation of 135 is 3 × 3 × 3 × 5

The prime factorisation of 225 is 3 × 3 × 5 × 5.

Hence, the prime factors that is common in both the numbers is 3 × 3 × 5

Therefore, the product of prime factors of 3 × 3 × 5 is 45.

Hence, the HCF of 135 and 225 is 45.

4. What is the HCF of 408 and 1032?

To find: HCF (408, 1032).

The prime factorisation of 408 is 2 × 2 × 2 × 3 × 17

The prime factorisation of 1032 is 2 × 2 × 2 × 3 × 43

Thus, the product of the prime factors of 2 × 2 × 2 × 3 is 24

Hence, the HCF of 408 and 1032 is 24.

5. Find the LCM of 3 and 4.

To find the LCM of 3 and 4 using the listing method, first list out the multiples of 3 and 4.

Thus, the multiples of 3 are 3, 6, 9, 12 , 15, 18, 21, 24, …

The multiples of 4 are 4, 8, 12 , 16, 20, 24, 28, …

From the above list, 12 is the smallest number that is common in both the multiples list.

Therefore, the LCM of 3 and 4 is 12.

6. What is the LCM of 6 and 8?

To find: LCM (6, 8).

The multiples of 6 are 6, 12, 18, 24 , 30, 36, 42, 48, …

The multiples of 8 are 8, 16, 24 , 32, 40, 48, ….

Thus, the smallest common multiple of 6 and 8 is 24.

Therefore, the LCM of 6 and 8 is 24.

7. Determine the LCM of 4 and 12 using the prime factorisation method.

To find the LCM of 4 and 12 using the prime factorisation method, follow the below steps.

Step 1: Find the prime factorization of given numbers:

The prime factorisation of 4 is 2 × 2

The prime factorisation of 12 is 2 × 2 × 3.

Step 2: The LCM of given numbers is found by multiplying the product of all factors. (Note: The common factor is included only once)

Hence, the product of prime factors = 2 × 2 × 3 = 12.

Therefore, the LCM of 4 and 12 is 12.

8. What is the LCM of 54 and 60?

The prime factorisation of 54 is 2 × 3 × 3 × 3.

The prime factorisation of 60 is 2 × 2 × 3 × 5.

Thus, the product of prime factors = 2 × 2 × 3 × 3 × 3 × 5 = 540

Hence, the LCM of 54 and 60 is 540.

9. Prove that LCM(12, 7) × HCF(12, 7) = Product(12, 7).

Given: LCM(12, 7) × HCF(12, 7) = Product(12, 7) …(1)

Finding LCM (12, 7):

The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84 , 91, 98, ..

The multiples of 12 are 12, 24, 36, 48, 60, 72, 84 , 96, 120, …

Hence, the LCM of 12 and 7 is 84.

Finding HCF (12, 7):

The factors of 7 are 1 and 7.

The factors of 12 are 1 , 2, 3, 4, 6 and 12.

Hence, HCF of 12 and 7 is 1.

Finding Product of 12 and 7:

The product of 12 and 7 = 12 × 7 = 84.

Now, substitute the obtained values in (1), we get

84 × 1 = 84

Hence, LHS = RHS

Therefore, LCM(12, 7) × HCF(12, 7) = Product(12, 7) is proved.

10. Find the LCM of the fractions 2/5, 4/7 and 6/11.

Given fractions: 2/5, 4/7 and 6/11

As we know, the formula to find the LCM of fractions is:

LCM of fractions = LCM of Numerators/HCF of Denominators .. (1)

Thus, the LCM of Numerators = LCM (2, 4, 6) = 12.

HCF of denominators = HCF (5, 7, 11) = 1

Now, substitute the values in (1), we get

LCM of fractions = 12/1 = 12

Hence, the LCM of the fractions 2/5, 4/7 and 6/11 is 12.

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Solve the following HCF and LCM Questions:

Find the LCM of 24 and 51.
What is the GCF of 68 and 101?
Find the LCM of fractions 11/6, 9/4, and 21/8.

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Word Problems on L.C.M.

Let us consider some of the word problems on l.c.m. (least common multiple).

1. Find the lowest number which is exactly divisible by 18 and 24.

We find the L.C.M. of 18 and 24 to get the required number.

$LCM Problems$

L.C.M. = 2 × 3 × 3 × 4 = 72

Therefore, 72 is the required number.

2. Find the lowest number which is less by 5 to be divided by 16, 24 and 36 exactly.

We find the L.C.M. of 16, 24 and 36.

$Word Problems on L.C.M.$

L.C.M. = 2 × 2 × 2 × 3 × 2 × 3 = 144

Now subtract 5 from 144 to get the required number.

144 - 5 = 139

Therefore, 139 is the required number.

3. Find the lowest number which is more by 6 to be divided by 25, 40 and 60 exactly.

We find the L.C.M. of 25, 40 and 60.

$L.C.M. of 25, 40 and 60$

L.C.M. = 2 × 2 × 5 × 5 × 2 × 3 = 600

Therefore, the required number is 600 + 6 = 606.

4. A shopkeeper sells candles in packets of 12 and candle stands in packet of 8. What is the least number of candles and candle stands Nita should buy so that there will be one candle for each candle stand.

To find a quantity which is the lowest common multiple of different quantities, we find the LCM.

Multiples of 12 are 12, 24, 36, 48, ……

Multiples of 8 are 8, 16, 24, 32, 40, ……

The lowest common multiple is 24. So, the least number of candles and candle stand that Nita should buy is 24.

$Word Problems on L.C.M.$

5. Find the lowest number which leaves 3 as remainder when divided by 8, 12 and 16.

We find the L.C.M. of 8, 12 and 16.

$Find the Lowest Number$

L.C.M. = 2 × 2 × 2 × 3 × 2 = 48

If we add 3 to 48 it becomes 51 which leaves 3 as remainder when divided by 8, 12 and 16.

Therefore, the required number is 48 + 3 = 51.

6. A florist wants to arrange 24 boquets of flowers in different rows. Find out in how many ways he can arrange the bouquets with same number in each row.

We need to find all the factors of 24.

24 = 1 × 24, 24 = 2 × 12, 24 = 3 × 8, 24 = 4 × 6

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24

He can arrange rows of 1, 2, 3, 4, 6, 8, 12 and 24 boquets.

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Least common multiple word problems

Eight interesting and fun least common multiple word problems you can give to your students to tease them. If your students can solve these problems, they can probably solve any word problems about the least common multiple.

Word problem #1 Today, both the soccer team and the basketball team had games. The soccer team plays every 3 days and the basketball team plays every 5 days. When will both teams have games on the same day again?

Word problem #2

A manager at a restaurant can buy hamburger buns in packages of 8 and hamburger patties in packages of 6. Suppose that the manager cannot buy part of a package. What is the least number of packages of each product he can buy to have an equal number of hamburger patties and buns?

Word problem #3

A man smiles at his beautiful wife every 3 seconds while the wife smiles back at him every 6 seconds. When will both husband and wife smile at each other at the same time?

Word problem #4

Steve can save 9 dollars every day while Maria can save 12 dollars every day. What is the least number of days it will take each person to save the same amount of money?

More Interesting and fun least common multiple word problems

Boxes that are 12 inches tall are being piled next to boxes that are 10 inches tall. What is the least height in feet at which the two piles will be the same height? Word problem #6 A radio station plays "yesterday" by the Beatles once every 2 days. Another radio station plays the same song once every 3 days. How many times in 30 days will both radio stations play the same song on the same day?

Word problem #7 Two men running a marathon took a sip of water at the same time 72 minutes after they started the race. If the first man took a sip of water every 9 minutes, how often did the other man take a sip of water? Word problem #8 A train to New York city leaves a station every 7 minutes. Another train to Boston leaves the station every 6 minutes. Suppose it is 6:30 am right now. At what time will both trains leave the station together?

A least common multiple word problem about barking dogs

Five dogs in a neighborhood were barking consistently last night. The names of the dogs are Lucy, Max, Murphy, Daisy, and Sam. The dogs started barking at 10 P.M. Then, Lucy barked every 5 minutes, Max barked every 3 minutes, Murphy barked every 4 minutes, Daisy barked every 3 minutes, and Sam barked every minute. Why did Mr. Smith suddenly awaken at 11 P.M.?

What is the least common multiple

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Factors and multiples.

These word problems need the use of greatest common factors (GCFs) or least common multiples (LCMs) to solve. Mixing GCF and LCM word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions.

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LEAST COMMON MULTIPLE WORD PROBLEMS

Problem 1 :

Omar is planting trees. He has enough trees to plant 6, 7, or 14 trees in each row. What is the least number of trees Omar could have ?

Solution :

To find the least number of trees, we have to find the least number that is evenly divisible by 6, 7 and 14. That is the least common multiple of 6, 7 and 14.

Find the least common multiple of 6, 7 and 14.

LCM = Product of all prime factors

= 7 ⋅ 2 ⋅ 3

The least number of trees required to plant 6, 7 or 14 tress in each row is 42.

Problem 2 :

The Line A bus arrives at the bus stop every 25 minutes, and the Line B bus arrives every 15 minutes. They are both at the bus stop right now. In how many minutes will they both be at the bus stop again ?

For example, let the two buses arrive at the bus stop after every 3 minutes and 4 minutes.

Then the Line A bus arrives after 3, 6, 9, 12 minutes......

Like this, the Line B bus arrives after 4, 8, 12, 16 minutes......

If both the buses arrive now, again they will arrive together after 12 minutes. This 12 is the least common multiple (LCM) of 3 and 4.

The same thing happened in our problem. To find the time, when both the buses are at the bus stop again, we have to find the LCM of (25, 15).

LCM (25, 15) = 75

75 minutes = 1 hour 15 minutes

So, both the buses will be at the bus stop again in 1 hour 15 minutes.

Problem 3 :

The high school marching band rehearses with either 6 or 10 members in every line. What is the least number of people that can be in the marching band ?

To find the least number of people, we have to find the least number that is evenly divisible by 6 and 10. That is the least common multiple of 6 and 10.

LCM (6, 10) = 30

The least number of people i n the marching band is 30.

Problem 4 :

Two numbers are in the ratio 4 : 7. If the second number is 35, find their least common multiple.

Because the two numbers are in the ratio 4 : 7, the numbers can be assumed as 4x and 7x.

But, it is given that the second number is 35.

Then,

7x = 35

Divide each side by 7.

x = 5

The first number = 4(5) = 20.

LCM (20, 35) = 140

So, the least common multiple of the two numbers is 140

Problem 5 :

Two numbers are in the ratio 2 : 3 and their least common multiple is 84 . Find the numbers.

Because the two numbers are in the ratio 2 : 3, the numbers can be assumed as 2x and 3x.

Least common multiple (2x, 3x) = 6x

But, it is given that the least common multiple of the two numbers is 84.

6x = 84

Divide each side by 6.

x = 14

Substitute x = 8 in 2x and 3x.

2x = 2(14) = 28

3x = 3(14) = 42

So, the two numbers are 28 and 42.

Problem 6 :

Two numbers are in the ratio 5 : 6 and their sum is 44. Find their least common multiple.

Because the two numbers are in the ratio 5 : 6, the numbers can be assumed as 5x and 6x.

5x + 6x = 44

11x = 44

Divide each side by 11.

x = 4

Substitute x = 4 in 5x and 6x.

5x = 5(4) = 20

6x = 6(4) = 24

The two numbers are 20 and 24.

LCM (20, 24) = 120

So, the least common multiple of the two numbers is 120.

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Word Problems on L.C.M | Least Common Multiples Word Problems with Answers

Enhance your math knowledge by practicing the Word Problems Involving Least Common Multiple for free of cost. LCM will lay a stronger foundation in learning all other similar math concepts as well increases your problem-solving ability. Refer to the Word Problems on LCM available here and get an idea of how to solve LCM Questions using various methods. Be familiar with the concept and attempt the exams on different questions framed on Least Common Multiple with utmost confidence and score well.

Also, Refer:

Word Problems on H.C.F
Worksheet on LCM
Worksheet on H.C.F

LCM Word Problems with Answers

Example 3. A shopkeeper sells chocolate Packets that contain 15 chocolate and biscuit packets that contain 9 biscuits. What is the least number of chocolates and biscuits Siri should buy so that there will be one biscuit for every chocolate? Solution: We have to find the LCM for Knowing how many chocolates and biscuits Siri should buy. Multiples of 15 are 15,30,45,60,75 Multiples of 9 are 9,18,27,36,45,54 The lowest common multiple is 45. The least number of chocolates and biscuits Siri should buy is 45.

Example 4. A teacher has to arrange 25 students in different rows. Find out how many ways he can arrange the students with the same number in each row. Solution: We have to find all the factors of 25. 25= 1× 25, 5 × 5 Factors of 25 are 1, 5, and 25. A Teacher can arrange students in rows of 1, 5,25.

Example 5. Pratima attends dance classes every 6 days, singing classes every 4 days, and yoga classes every 16 days. If she does all three activities today, in how many days will she do all the three activities again on the same day? Solution: Multiples of 4 are 4,8,12,16,20,24,28,32,36,40,44,48,52 Multiples of 6 are 6,12,18,24,30,36,42,48,54 Multiples of 16 are 16,32,48 In 48 days she will do all three on the same day.

Example 6. Nevada goes shopping every 30 days. If she did shopping on Friday, find how many days she has to wait for shopping again on a Friday? Solution: For finding a common multiple we have to multiply 30 days by 7 days in a week(to end up Friday again) 30 × 7=210. Hence, Nevada has to wait for 210 days for shopping.

Example 8. Find the lowest number to be more by 5 that is exactly divisible by 40,60,80.

Example 9. A shopkeeper sells tea shirts 24 and pants 12. What is the least number of pants and shirts Pranavi should buy so that there will be one shirt for each pair of pant? Solution: To find the least number of pants and shirts we find LCM. Multiples of 24 are 24,48,72,96,120…. Multiples of 12 are 12,24,36,48,60…… The least common multiple is 24. Hence, The least number of pants and shirts Pranavi should buy is 24.

Example 10. Jagadish has to arrange 30 brinjal plants in different rows. Find out how many ways he can arrange the brinjal plants with the same number in each row. Solution: We have to find all the factors of 30. 30= 1× 30, 2 ×15, 3 × 10, 5 × 6 Factors of 30 are 1,2,3,5,6,10, 15 and 30. Jagadish can arrange brinjal plants in rows of 1,2,3,5,6,10,15,30.

Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

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People hold plastic water containers, waiting to fill them at a tap outside a store.

India’s Silicon Valley Faces a Water Crisis That Software Cannot Solve

Bengaluru gets plenty of rain. But the city did not properly adapt as its soaring population strained traditional water sources.

Filling up with subsidized water at a government distribution center in the Indian city of Bengaluru, also known as Bangalore. Credit...

Supported by

By Damien Cave

Photographs by Atul Loke

Reporting from Bengaluru, India

Published March 31, 2024 Updated April 2, 2024

The water tankers seeking to fill their bellies bounced past the dry lakes of India’s booming technology capital. Their bleary-eyed drivers waited in line to suck what they could from wells dug a mile deep into dusty lots between app offices and apartment towers named for bougainvillea — all built before sewage and water lines could reach them.

At one well, where neighbors lamented the loss of a mango grove, a handwritten logbook listed the water runs of a crisis: 3:15 and 4:10 one morning; 12:58, 2:27 and 3:29 the next.

“I get 50 calls a day,” said Prakash Chudegowda, a tanker driver in south Bengaluru, also known as Bangalore, as he connected a hose to the well. “I can only get to 15.”

People fill weathered plastic buckets with water that is coming out of a series of taps, lined up in a row.

The Silicon Valley of South Asia has a nature issue — a pain point that software cannot solve. In the sprawl beyond Bengaluru’s core, where dreams of tech riches usually grow, schools lack water to flush toilets. Washing machines have gone quiet. Showers are being postponed, and children with only dirty water to drink are being hospitalized with typhoid fever.

The big problem afflicting Bengaluru is not a lack of rain (it gets plenty, about as much as Seattle), but rather what often holds this giant, energetic nation back: arthritic governance. As the city rushed toward the digital future, tripling its population to 15 million since the 1990s and building a lively tech ecosystem, water management fell behind and never caught up as otherwise healthy aquifers were drawn dry by the unchecked spread of urban bore wells.

Failures of environmental stewardship are common across a country with severe pollution and an acute need for economic growth to provide for 1.4 billion people, spanning political parties and India’s north-south divide . But Bengaluru’s water struggle is especially withering for many — and motivating for some who have water sales or reform in mind — because the city sees itself as an innovator. And in this case, the causes and solutions are well known.

“There is no crisis of water availability,” said Vishwanath Srikantaiah, a water researcher and urban planner in Bengaluru. “It’s a clear-cut crisis of state failure.”

Viewed another way, he added in an interview at his home, where books about water and rivers were stacked nearly to the ceiling, it is a crisis caused by a lack of imagination.

As public policy experts tell it, Bengaluru and the broader state of Karnataka have been too slow to plan for growth, too divided across agencies and too rigid in their reliance on pumping water uphill from reservoirs along the Kaveri River more than 50 miles away.

Despite a long history of local hydrology — Nadaprabhu Kempegowda, the 16th-century founder of Bengaluru, built hundreds of cascading lakes for irrigation — officials have mostly stuck with the traditional engineering option that their predecessors turned to in the 1950s and ’60s.

That is the case despite its challenges and expense. The energy cost alone for pumping eats up 75 percent of the Bangalore Water Supply and Sewerage Board’s revenue, while supplying only around half of what the city needs.

The rest, for decades, has come from bore wells — holes about six inches wide that act like straws for water from aquifers below. An authority separate from the water board has punched 14,000 of them into the ground, half of which are now dry, according to officials. Experts estimate that residents have drilled an additional 450,000 to 500,000 into the cityscape, without the government’s knowing where or having a clear sense of their impact.

In much of the city, the wells are like doorbells, plentiful but seemingly invisible until someone points them out. Drilling failures appear as cutout circles on quieter streets; successes are often covered in flowers, with a black hose snaking into a home down the street.

Spending a day in the cab of Mr. Chudegowda’s tanker truck offered a glimpse of how the ad hoc system works. At one stop, drivers wrote their times in a logbook while cameras watched how much they took. At another the supply was slow and organized: A half-dozen drivers took 20-minute turns for fill-ups of around 6,000 liters, or about 1,600 gallons, just a few steps from a lake depleted to a puddle. At a third, a building owner sold a load to Mr. Chudegowda without the wait.

“Every minute counts,” he said as he climbed out of the truck.

His customers ranged from a bra factory with 100 workers to a small apartment building, all within a few miles to maximize profit. He charged each up to 1,500 rupees ($18) for each tanker load, more than double the going rate from a few months ago, which he considered justified because costs had gone up.

Drills — easily hired from companies with storefronts across the city — often fail to find water or have to go deeper now, which means more electricity and gas for the pumps pulling precious liquid from the earth.

The effects, while not at “Dune”-like levels, have become more visible in recent weeks, especially in the tech corridors, with their blur of luxury apartments, slums, mobile phone stores, malls, in vitro fertilization clinics and shimmering offices.

In Whitefield, a busy software hub, Sumedha Rao, a teacher at a new public school, offered to ask her class of 12-year-olds about their experiences with water scarcity. The hallways were painted in bright colors with words of encouragement — resilience, citizenship, collaboration. In class, they were asked how often they have water at home.

“One day a week, ma’am,” said a girl with pigtails.

“We just have a bucket,” said a boy near the back.

“There’s no water in the bore wells,” shouted another.

Many take small amounts of drinking water from school taps for their families — only one water bottle per child, because it is all the school can spare. Behind a play area the color and consistency of ground ginger sat a hulking pile of metal: a broken bore well.

“The motor stopped working,” said Shekar Venkataswamy, a physical education teacher with a brigand’s mustache.

Walking toward his home behind the school, he pointed to a dry hole where drilling failed, and one where it worked. A few thousand families take turns using the water for an hour each, with an elaborate schedule that is tightly managed.

Community leaders expressed pride in how they were handling the crisis, softening the blows of sacrifice. Many others have been inspired to broader action.

One morning, four tech workers who had become water activists showed up in a northern corner of the city where Mr. Srikantaiah, the water researcher, had worked with the local community to rejuvenate a once trash-strewn lake. A small network of gurgling filters and pipes sends out 200,000 liters of potable water per day.

“It will soon be 600,000,” Mr. Srikantaiah said. And the price per customer: nearly a third of what tanker drivers are charging.

The tech workers said they planned to share the details with neighbors and officials, to spread the word that a lake, using rainwater and lightly treated sewage, could be turned into a safe, affordable, reliable water source.

In an interview at his office, the chairman of the water board, Ram Prasath Manohar, 43, a seasoned government administrator installed three months ago, embraced the idea.

Acknowledging that some past officials had thought narrowly about water management, he said he hoped to attract public and private money for a more innovative approach, mixing data-driven methods that would revive lakes to let aquifers recharge and would expand rainwater harvesting and conservation.

“We’re going for a greener solution,” he said. “A more effective solution.”

So far, though, progress has been slow. He has not been able to hire any additional staff, he said, and he is working from 6 a.m. to 2 a.m. every day.

Short-term relief, he prays, will come in the next few weeks, with reservoir water extended to more parts of the city and the expected spring rains. Most of all, like many others in India’s Silicon Valley, he hopes all the public attention to water scarcity will add momentum for long-term change.

In one corner of his offices, a quote from Benjamin Franklin had been printed on a piece of paper and pasted to a window: “When the well is dry, we know the worth of water.”

“This crisis,” he said, rubbing his tired eyes, “it gives us an opportunity.”

Imran Khan Pathan contributed reporting.

Damien Cave is an international correspondent for The Times, covering the Indo-Pacific region. He is based in Sydney, Australia. More about Damien Cave

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The 3-body problem is real, and it’s really unsolvable

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Rosalind Chao as Ye Wenjie standing in the middle of three overlapping circles

Everybody seems to be talking about 3 Body Problem , the new Netflix series based on Cixin Liu’s Remembrance of Earth’s Past book trilogy . Fewer people are talking about the two series’ namesake: The unsolvable physics problem of the same name.

This makes sense, because it’s confusing . In physics, the three-body problem attempts to find a way to predict the movements of three objects whose gravity interacts with each of the others — like three stars that are close together in space. Sounds simple enough, right? Yet I myself recently pulled up the Wikipedia article on the three-body problem and closed the tab in the same manner that a person might stagger away from a bright light. Apparently the Earth, sun, and moon are a three-body system? Are you telling me we don’t know how the moon moves ? Scientists have published multiple solutions for the three-body problem? Are you telling me Cixin Liu’s books are out of date?

All I’d wanted to know was why the problem was considered unsolvable, and now memories of my one semester of high school physics were swimming before my eyes like so many glowing doom numbers. However, despite my pains, I have readied several ways that we non-physicists can be confident that the three-body problem is, in fact, unsolvable.

Reason 1: This is a special definition of ‘unsolvable’

Jin Cheng (Jess Hong) holds up an apple in a medieval hall in 3 Body Problem.

The three-body problem is extra confusing, because scientists are seemingly constantly finding new solutions to the three-body problem! They just don’t mean a one-solution-for-all solution. Such a formula does exist for a two-body system, and apparently Isaac Newton figured it out in 1687 . But systems with more than two bodies are, according to physicists, too chaotic (i.e., not in the sense of a child’s messy bedroom, but in the sense of “chaos theory”) to be corralled by a single solution.

When physicists say they have a new solution to the three-body problem, they mean that they’ve found a specific solution for three-body systems that have certain theoretical parameters. Don’t ask me to explain those parameters, because they’re all things like “the three masses are collinear at each instant” or “a zero angular momentum solution with three equal masses moving around a figure-eight shape.” But basically: By narrowing the focus of the problem to certain arrangements of three-body systems, physicists have been able to derive formulas that predict the movements of some of them, like in our solar system. The mass of the Earth and the sun create a “ restricted three-body problem ,” where a less-big body (in this case, the moon) moves under the influence of two massive ones (the Earth and the sun).

What physicists mean when they say the three-body problem has no solution is simply that there isn’t a one-formula-fits-all solution to every way that the gravity of three objects might cause those objects to move — which is exactly what Three-Body Problem bases its whole premise on.

Reason 2: 3 Body Problem picked an unsolved three-body system on purpose

A woman floating in front of three celestial bodies (ahem) in 3 Body Problem

Henri Poincaré’s research into a general solution to the three-body problem formed the basis of what would become known as chaos theory (you might know it from its co-starring role in Jurassic Park ). And 3 Body Problem itself isn’t about any old three-body system. It’s specifically about an extremely chaotic three-body system, the exact kind of arrangement of bodies that Poincaré was focused on when he showed that the problem is “unsolvable.”

[ Ed. note: The rest of this section includes some spoilers for 3 Body Problem .]

In both Liu’s books and Netflix’s 3 Body Problem , humanity faces an invasion by aliens (called Trisolarans in the English translation of the books, and San-Ti in the TV series) whose home solar system features three suns in a chaotic three-body relationship. It is a world where, unlike ours, the heavens are fundamentally unpredictable. Periods of icy cold give way to searing heat that give way to swings in gravity that turn into temporary reprieves that can never be trusted. The unpredictable nature of the San-Ti environment is the source of every detail of their physicality, their philosophy, and their desire to claim Earth for their own.

In other words, 3 Body Problem ’s three-body problem is unsolvable because Liu wanted to write a story with an unsolvable three-body system, so he chose one of the three-body systems for which we have not discovered a solution, and might never.

Reason 3: Scientists are still working on the three-body problem

Perhaps the best reason I can give you to believe that the three-body problem is real, and is really unsolvable, is that some scientists published a whole set of new solutions for specific three-body systems very recently .

If physicists are still working on the three-body problem, we can safely assume that it has not been solved. Scientists, after all, are the real experts. And I am definitely not.

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Quantum Physics

Title: solving the $kp$ problem with the global cartan decomposition.

Abstract: Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called $KP$-problem, where target unitaries belong to a semi-simple Lie group manifold $G$ whose Lie algebra admits a $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in $\mathfrak{p}$. In this paper, we propose a new method utilising global Cartan decompositions $G=KAK$ of symmetric spaces $G/K$ for generating time-optimal unitaries for targets $-iX \in [\frak{p},\frak{p}] \subset \frak{k}$ with controls $-iH(t) \in \frak{p}$. Target unitaries are parametrised as $U=kac$ where $k,c \in K$ and $a = e^{i\Theta}$ with $\Theta \in \frak{a}$. We show that the assumption of $d\Theta=0$ equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by $\mathfrak{p}$ and local rotations are generated by $[\mathfrak{p},\mathfrak{p}]$.

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How Much Longer Can Netanyahu Hold On?

His coalition has held through the war, but a shortage of military manpower is exposing fissures.

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Benjamin Netanyahu has been able to count on the support from Israel’s ultra-Orthodox parties. But as the IDF needs additional manpower to continue a two-front war, the military exemption that the ultra-Orthodox enjoy is coming under scrutiny, and could fracture Netanyahu’s hold on power.

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Nitrate has plagued Wisconsin's groundwater for decades. Why is the problem so hard to solve?

CASCO, Wis. - When Tyler Frye purchased his home in 2020, he saw the Casco area in Kewaunee County as the perfect place for him and his fiancée to start a family. The house was new, in a just-built subdivision close to his family and the area where he grew up. He snapped it up.

Then he discovered nitrate contamination in the water.

"When I bought the house, I falsely assumed with the new well, the water would be safe," he said.

Since then, he and his fiancée, Shannon, have married, but the question over what to do about the water remains. Frye gets angry if he thinks about it too long, he said. And the couple is worried that if they have children in the house, it could put them at risk for health issues. Drinking water with elevated nitrate levels poses special risks to infants and is being increasingly linked to cancers and other diseases in adults.

"It was pretty devastating," he said. "Owning a home is the American dream. I went to college, got a degree, got a job. I sacrificed a lot to make money to buy a nicer house."

The contamination makes the worth less than Frye paid for it, so selling would be a financial hit. But keeping it means maintaining a filtration system indefinitely to lower the levels of nitrate in the water, but it still wouldn't reduce nitrate to nondetectable levels.

Frye bought a $1,000 reverse osmosis system, which brought the nitrate levels down from 17.4 parts per million to 3 ppm, but he's leery of having any detectable amounts of nitrate in his water.

"It's unsettling," Frye said. "And we talked about once we start having kids, we're not going to trust it to provide water for our kids."

Nitrate is the state’s most widespread contaminant of groundwater. About 10%, or 80,000, of the state’s private wells fail to meet the state drinking water standard of 10 mg/L. In heavily farmed areas, that percentage can rise upwards of 20%, according to a 2023 report to the Legislature from the state's Groundwater Coordinating Council.

The contaminant mostly comes from manure or fertilizer but can also originate from septic systems and other sources.

In the 2023 report, members of the Groundwater Coordinating Council noted that nitrate contamination has been impacting Wisconsin waters for more than 50 years. That decades-long struggle, with not much to show for its efforts, has left people fatigued and apathetic, especially as new water quality concerns like PFAS continue to emerge. PFAS, also known as "forever chemicals," are man-man compounds that are used in products like firefighting foam and water-proof clothing that are known to last for decades in the environment.

While tighter rules have been proposed, Wisconsin has largely lost steam in its regulation of nitrate, leaving more headaches for private well owners across the state. To some, the issue feels insurmountable. Some don't even want to test their water, for fear of unearthing the problem.

"Nitrate is really hard for people," Jen McNelly, formerly Portage County's water resource specialist, said in a December 2023 interview. McNelly has since departed for a position at UW-Madison’s Division of Extension. "People get exhausted talking about it, to a certain extent, because there is no easy answer and there’s no silver bullet."

Complicating matters is the role that agriculture — a critical part of Wisconsin's economy, history and culture — plays in nitrate contamination. Some farmers are reluctant to change practices unless homeowners are also willing to examine their septic systems. Others acknowledge the negative impact of nitrogen fertilizers but are hamstrung by market forces that consistently demand bigger crop yields.

Wisconsin is far from the only state struggling to remedy nitrate contamination. In Minnesota, the EPA has told the state to take more action to protect its residents . In Nebraska, owners of private wells are being urged to test their private wells to gather more data on the extent of nitrate contamination. In Oregon, private companies are offering to chip in to help with nitrate pollution , after more than 30 years of struggles.

As time goes on, though, more Wisconsinites like Frye are unable to freely drink the water that comes out of their tap. And they may not have another 50 years to wait for a solution.

Nitrate's health risks pose worries

Nitrate forms when oxygen combines with nitrogen-rich sources like fertilizer, manure or waste from septic tanks. It slips easily into groundwater by way of rain or melting snow, especially in places with shallow soil or fragmented bedrock.

1994 research from the University of Wisconsin-Stevens Point found that 90% of nitrogen inputs to groundwater can be traced to agriculture, with the rest coming from septic systems and lawn fertilizer. Kevin Masarik, a groundwater education specialist at UW-Stevens Point, said the breakdown of sources likely has not changed since then.

Farming remains a crucial part of Wisconsin’s economy, and applying nitrogen fertilizer can increase crop yields. The problem arises when more nitrogen is applied to the ground than the plants can take up, which means it'll get into the groundwater somehow, said Steve Elmore, program director of the Wisconsin Department of Natural Resources' Bureau of Drinking Water and Groundwater.

Since farmers want to maximize yield, they may add extra fertilizer as "insurance," Elmore said. But that's not the only reason. If heavy rain falls after a farmer has recently laid down fertilizer, it can be washed away.

Even if fertilizer is applied with great care toward the timing and amount, it could still leach too much nitrogen into groundwater. That's because it just takes a lot of nutrients to grow row crops, Elmore said.

Large farms, or concentrated animal feeding operations, are also a source of nitrate contamination, because of the large amounts of manure produced by the animals. The manure is typically mixed with water and stored in tanks or lagoons, then spread on farm fields, where the nitrate can sometimes run off into streams, rivers and lakes, or soak down through the soil and into groundwater.

Nitrogen can be damaging to the environment when it's washed into surface water, driving excessive algae growth and causing harm to fish and other aquatic life. But there are dangerous consequences if too much ends up in groundwater — the source of nearly 75% of Wisconsin's drinking water.

The U.S. Environmental Protection Agency sets the maximum contaminant level for nitrate , or the amount that's considered safe for drinking water, at 10 mg/L. That level was set to protect infants from blue baby syndrome , which occurs when there's not enough oxygen in a baby's blood and can be caused by high nitrate levels in drinking water.

There's been less study of what elevated nitrate can do to adults, but some research is beginning to show broader health impacts. A 2018 review of studies of the health impacts of nitrate found that risk of illnesses like colorectal cancer, thyroid disease and neural tube defects increased even when people were ingesting nitrate in their drinking water below the maximum contaminant level.

Even though the health risks may not yet be fully understood, people are beginning to worry. Kriss Marion, a Lafayette County supervisor and clean water advocate, said she attends a lot of funerals, not unusual for an aging community. What is unusual is the number of young people in the area getting cancer and dying, too. Lafayette County is in south-central Wisconsin, one of the state's hotspots for nitrate contamination.

"A lot of conversation at funerals is about water — whether water was the reason," Marion said.

Wisconsin's nitrate problem is growing

Nitrate trends are hard to tease out. One-third of private well owners in Wisconsin have never had their well tested for nitrate, and even private wells that have been tested aren't usually sampled consistently over time. Further, extreme weather swings wrought by climate change , like floods and droughts, cause nitrate levels in groundwater to fluctuate.

But the data that are available show — and state water quality experts agree — Wisconsin is losing ground on its nitrate problem.

One way to assess nitrate trends is by looking at the levels in surface water. A large chunk of the state is covered by "groundwater-dominated" watersheds — in other words, streams and rivers that flow through the area are resupplied by groundwater. Nitrate is increasing at most of the sites where the DNR monitors surface water quality.

Elmore acknowledged that Wisconsin continues to struggle with nitrate contamination, especially in comparison to its handling of another common component of fertilizer, phosphorus. In 2010, Wisconsin enacted some of the most stringent regulations in the country to address phosphorus. DNR surface water quality monitoring shows concentrations of phosphorus are decreasing in most rivers across Wisconsin.

The DNR first launched the rulemaking process for standards regarding nitrates in groundwater in December 2019 in a bid to keep the nutrients from affecting soils and geology most vulnerable to contamination. The rulemaking process typically takes 30 months and then requires signoff from the Legislature. But the rule had to be abandoned in 2021, due to the high estimated cost to businesses.

Outside of setting rules, encouraging farmers to take up practices that protect soil health can decrease nitrogen leaching. Elmore said cover cropping, which covers and enriches the soil outside the growing season, and no-till farming, which minimizes soil disturbance, can be useful.

But making a wholesale transition to those practices can be costly to farmers. If people want to keep their drinking water safe, they could expect to pay more at the grocery store for food that was cultivated that way, Masarik suggested — similar to higher-priced grass-fed beef or organic products.

'We're not attacking farming'

Kewaunee County, where Frye lives, has become a flashpoint in Wisconsin's debate over nitrate contamination.

The karst geography, which consists of rock that easily allows contamination from the soil to sink into the groundwater, makes the area more susceptible than most other areas of the state. And with a growing concentration of mega-farms that spread large amounts of liquified manure each year, the problem isn't getting any better, despite residents' outcries.

Many of the homeowners in the county have seen contamination in their wells, requiring filtration or routine sanitation. Rhonda Spude, a resident of Algoma who grew up in the area, has had consistent levels of nitrate and bacteria in her well for years. While the nitrate level is hovering just under the state's limits, the reoccurring bacteria makes her water unsafe to drink without boiling.

For the most part, her family now just uses bottled water, which they pay for out of their own pockets.

It's frustrating because growing up in the area, there weren't any factory farms, and her family never had issues with water quality. And from Spude's perspective, it seems like the county and the state aren't doing anything to make the situation better for her and her neighbors.

"They go round and round, and it's like nobody wants to take the blame for anything," she said. "They tried to say it's not anything that's that bad. Well have you had that problem? We've been dealing with it for years."

Nancy Utesch, a resident of the city of Kewaunee, and her husband have been working on the issue for years, pushing for solutions and regulations to protect the groundwater that so many residents rely on for drinking, cooking and bathing.

The work she does to inform the community of the risks tied to water contamination constantly receives criticism, because the farms in her area produce large amounts of food.

"At a local meeting, somebody said 'You shouldn't complain with a mouthful of food.' And that's where the conversation tends to go," she said. "But we're not attacking farming. My husband and I are farmers. We own 150 acres here, we own beef cattle and have a rotational grazing system and we understand farming. But we have to find a way where farming does not mean destroying our water."

One of the largest factory farms in Kewaunee, Kinnard Farms, has faced issues related to the amount of manure spread across the county. The farm was ordered to cap the number of cows it kept, and to install monitoring wells on fields that it spread on to monitor the groundwater, following a settlement with the state. The farm last year agreed to begin treating liquid manure instead of applying it to fields , in order to avoid having to install the monitoring wells, which the farm said could become costly.

But even with the settlements, there has been contention over the high concentration of factory farms in Kewaunee, and the growing number of private wells impacted by nitrate.

When residents pointed out consistent harm, blame was shifted elsewhere – to septic systems that had to be replaced.

But the contamination still lingers.

Utesch said the thought that pollution needs to be excused because farms are providing food isn’t correct.

"It's not a good enough thing to say, 'We're feeding you so your water is now going to be poisoned,'" she said.

"What we need is a farm model that seeks the idea of growing nutrient dense food that is actually healthy for people, for the environment and cares about the water. As long as we are hell-bent on growing corn and soy, we're going to have nitrate problems."

'There is no easy answer'

In Portage County, McNelly described nitrate as "by far our single biggest drinking water issue." She estimated 22% of the county's private wells exceed the drinking water standard for nitrate.

Located in Wisconsin's Central Sands region, named for its sandy soils, Portage County is a top potato producer . Nitrogen is critical to growing potatoes and getting high yields. The advent of irrigated agriculture also made a "huge change" to the productivity of the county's farmland, McNelly said — but irrigating fields with water that's high in nitrate and adding fertilizer on top worsens contamination.

County officials began to worry about nitrate in the 1970s and 1980s, McNelly said, leading to one of the state's first groundwater management plans. But they're still struggling with it today.

While some farmers are working on reducing the amount of nitrogen that goes onto their fields, the hard evidence on nitrogen reduction practices is still being developed, she said.

In the meantime, she said she wanted people whose drinking water is polluted to be able to use it again. McNelly and others have criticized the limitations of Wisconsin's well compensation program, which offers funding to replace a contaminated well but requires that a person have livestock and a nitrate level of 40 mg/L — four times the safe drinking water standard — to be able to access the funds.

The state used ARPA funding to launch a similar program open to anyone, which McNelly said has been used "heavily" in Portage County, but when she spoke with the Journal Sentinel in December, she'd recently received an email saying the money was running out. Elmore, with the DNR, called the $10 million in ARPA funding "a very small dent" in the estimated $450 million that would be needed to replace all the state's contaminated wells.

Katy Bailey, who bought what she and her husband thought would be their forever home in Nelsonville in 2012, used those well compensation funds to install a new well before the couple sold their house last July and moved to Wisconsin Rapids. Bailey and other Nelsonville residents were profiled in a Wausau Daily Herald story in 2020 about their fears that manure from a CAFO about a mile from the small Portage County community was leaching nitrogen into their groundwater.

Bailey, her husband and her young son all have thyroid issues, one symptom health officials say may be caused by continually consuming high levels of nitrate. It was the birth of her son that Bailey said caused her to think seriously about moving away from the village that she had grown to love.

"I really do think it was the discovery of the high nitrates that really fractured the whole integrity of the community," she said.

'What do we do now?'

In south-central Wisconsin, home to some of the most heavily farmed land in the state , a study meant to assess groundwater contamination momentarily exploded into the spotlight, but has since faded into the background.

The Southwest Wisconsin Groundwater and Geology Study , or SWIGG, as it came to be known, evaluated 840 samples from residents’ private drinking wells in Grant, Iowa and Lafayette counties for nitrate, total coliform bacteria and E. coli. Some mid-study results drew fierce backlash in 2019 when news reports misconstrued a statistic showing 91% of a sample of contaminated wells contained fecal matter.

But the study’s final results , published in 2022, have languished even though they show that the nitrate contamination in wells is severe.

In Lafayette County, for example, 27% of wells tested in November 2018 and 20% tested in April 2019 had unsafe nitrate concentrations.

The researchers found that the more acres of agricultural land near a well, the more likely the well water was to have elevated nitrate levels. No such association was found with proximity to septic systems, said Joel Stokdyk, a biologist with the U.S. Geological Survey who co-authored the study.

But fecal contamination in the samples came from both humans and animals, casting blame on septic systems.

“When (the results) did come out, they were sort of confusing enough that it made it hard for people to rally around any particular take or response,” said Marion, the Lafayette County supervisor.

That's clear in the variety of actions county officials have since tried to pursue to improve the area's groundwater. Erica Sauer, Lafayette County’s conservation, planning and zoning manager, has pushed to require manure storage facilities in the county to be constructed to minimize leakage and helped residents seal up abandoned wells so pollution can’t continue to seep into the ground. The county’s health department also set up a water testing lab for people to regularly test their well water.

Erik Heagle – Grant County’s conservation, sanitation and zoning administrator – wrote in an email that his department has focused on enforcement of septic system compliance and revamping its Farmland Preservation Program to have stricter standards on nutrient and manure spreading.

But he reported waning attention to the issue from the public.

“There is a willingness to address the issue,” Heagle wrote, “it just doesn’t seem like there is a sense of urgency about it.”

While Marion said people with polluted wells were showing up in droves to demand answers at county board meetings when the first round of study results dropped in 2019, both the COVID-19 pandemic and the rise of other pressing environmental issues, like PFAS, have deflated most of that pressure.

Most of all, she hopes to see more people encouraged to regularly test their well, even though they may hope their water is perfectly fine.

"We know a lot more than we did before," she said, "but science is only as powerful as your willingness to use it."

Scott Laeser, a Lafayette County resident, organic vegetable farmer and senior working lands advisor for the Rural Climate Partnership, said he felt the nitrate findings were eclipsed by the findings about fecal contamination.

Pointing fingers about who’s more to blame for pollution distracts from the science that clearly shows a problem with nitrate contamination, one that he feels has a clear major source: agriculture. Eventually, he said, the state will have to interrogate what it grows and how it farms.

Some of that work is taking place right in Lafayette County. In 2023, Wisconsin's Department of Agriculture, Trade and Consumer Protection dispersed its first grants for the Nitrogen Optimization Pilot Program to farmers across the state interested in collecting more data about their nitrogen fertilizer use.

Mike Berget, whose family grows over 10,000 acres of corn and beans and produces weaner pigs near Wiota, is one of those farmers. He's part of his local farmer-led watershed group — the Lafayette Ag Stewardship Alliance, or LASA — and he and three other farmers in the group will be testing different nitrogen rates on a rye cover crop to determine the amount that can produce a good corn yield but not leach into the soil.

LASA paid for Lafayette County's portion of the SWIGG study because farmers wanted to know what was going on, Berget said. He believes many of the tainted wells were contaminated decades ago, when farming practices were different, and thinks high fertilizer prices are causing farmers to apply less of it today.

Cutting down on nitrogen now to save money may lead farmers to stick with that reduced application, Berget said, if they see that they were able to get just as good a yield.

"Sure, we're not going to correct things overnight," he said, "but we've got to start somewhere."

Laeser would like to see more urgency from people tasked with correcting things.

"We're spinning our wheels on pollution when we recognize it's going to take time to fix this issue," Laeser said. "The people that are really getting the shaft are the ones who have contaminated water right now, and aren't getting the help they deserve."

Laura Schulte can be reached at [email protected] and on Twitter at @SchulteLaura . Madeline Heim is a Report for America corps reporter who writes about environmental issues in the Mississippi River watershed and across Wisconsin. Contact her at [email protected] .

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Across Duke, administrative professionals in schools and departments help their units function smoothly.

And when things get busy, plans change quickly, or problems arise, these colleagues often find themselves in the middle of the solution.

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Lowest Common Multiple
The lowest common multiple (LCM) of a finite set of non-zero integers is the smallest positive number that is a multiple of each integer in the set. It is a fundamental concept in number theory, and is closely related to the greatest common divisor.It is taught to elementary school students as an aid in adding fractions: to simplify $ \frac{a}{x}+\frac{b}{y}, $ one first writes the fractions ...
LCM Of Two Numbers Practice Problems With Answers
There are ten (10) practice problems here about how to find the Least Common Multiple (LCM) of Two Numbers. You can use any methods to find the answer such as the prime factorization method or list method. It's up to you which one to utilize as long as you arrive at the correct answer. ... Problem 4: Find the LCM of [latex]14[/latex] and ...
GCF & LCM word problems (video)
And that least common multiple is equal to 120. Now there's other ways that you can find the least common multiple other than just looking at the multiples like this. You could look at it through prime factorization. 30 is 2 times 15, which is 3 times 5. So we could say that 30 is equal to 2 times 3 times 5.
Least Common Multiple (LCM)
let's solve for the LCM of 15 and 20: First, find the Greatest Common Divisor (GCD) of 15 and 20. The GCD of 15 and 20 is 5. Then, multiply the numbers together: 15×20=300. Finally, divide the product by the GCD: 300 ÷ 5= 60. Therefore, the Least Common Multiple (LCM) of 15 and 20 is 60.
Least common multiple (video)
The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. To find the LCM, you can use the prime factorization method or list the multiples of each number. Prime factorization involves breaking down numbers into their prime factors and constructing the smallest number with all the factors.
Least Common Multiple (Simple How-To w/ 9+ Examples!)
The least common multiple, or LCM, is the smallest positive multiple that is common to two or more numbers. Find out more with 9+ step-by-step examples! Calcworkshop. ... whereas the prime factorization method is tried and true and will work for any integer as noted on the Art of Problem Solving. Example — Hard LCM Problem.
Least Common Multiple, LCM: Problems with Solutions
If x is the least common multiple of 8 and 9, find x. Problem 6. Find the least common multiple of 7 and 12. Problem 7. Find the least common multiple of 15 and 5. Problem 8. Find the least common multiple of 4 and 10. Problem 9. Find the least common multiple of 3 and 9.
LCM (Least Common Multiple) in Maths
Least Common Multiple (LCM) In Mathematics, the LCM of any two is the value that is evenly divisible by the two given numbers. The full form of LCM is Least Common Multiple. It is also called the Least Common Divisor (LCD). For example, LCM (4, 5) = 20. Here, the LCM 20 is divisible by both 4 and 5 such that 4 and 5 are called the divisors of 20.
GCF & LCM
The common factors of 12 and 18 are 1, 2, 3 and 6. The greatest number in these common factors is 6, hence the GCF of 12 and 18 is 6. LCM of 12 and 18 : The first four multiples of 12 are: 12, 24 ...
Least Common Multiple (LCM) Worksheets
Least common multiple worksheets provide exercises on finding the LCM of a set of numbers using common multiples and factoring technique. ... Enjoy an abundance of GCF and LCM real-world scenarios that test your problem-solving skills and broaden your learning. Read the problems, analyze the data, and apply your skills to find the GCD and HCF ...
HCF and LCM Questions with Solutions (Complete Explanations)
The Least Common Multiple (LCM) of 2 or more given numbers is defined as the smallest common multiple, which is found in both the numbers. ... Solve the following HCF and LCM Questions: Find the LCM of 24 and 51. What is the GCF of 68 and 101? Find the LCM of fractions 11/6, 9/4, and 21/8. Download BYJU'S - The Learning App today and learn ...
L.C.M. Word Problems
Word Problems on L.C.M. Let us consider some of the word problems on l.c.m. (least common multiple). 1. Find the lowest number which is exactly divisible by 18 and 24. Solution: We find the L.C.M. of 18 and 24 to get the required number. L.C.M. = 2 × 3 × 3 × 4 = 72. Therefore, 72 is the required number. 2.
Lowest Common Multiples (LCM) and Highest Common Factors ...
Click here for Answers. Practice Questions. Previous: Midpoint of Two Numbers Practice Questions. Next: Place Value Practice Questions. The Corbettmaths Practice Questions on the LCM and HCF.
Least Common Multiple Word Problems
If your students can solve these problems, they can probably solve any word problems about the least common multiple. Word problem #1 Today, both the soccer team and the basketball team had games. The soccer team plays every 3 days and the basketball team plays every 5 days. ... A least common multiple word problem about barking dogs. Five dogs ...
GCF and LCM Word Problems
These word problems need the use of greatest common factors (GCFs) or least common multiples (LCMs) to solve. Mixing GCF and LCM word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions. Worksheet #1 Worksheet #2 Worksheet #3 Worksheet #4. Worksheet #5 Worksheet #6. Similar:
Least Common Multiple Word Problems
x = 5. The first number = 4 (5) = 20. LCM (20, 35) = 140. So, the least common multiple of the two numbers is 140. Problem 5 : Two numbers are in the ratio 2 : 3 and their least common multiple is 84. Find the numbers. Solution : Because the two numbers are in the ratio 2 : 3, the numbers can be assumed as 2x and 3x.
GCF & LCM word problems (practice)
GCF & LCM word problems. Google Classroom. You might need: Calculator. Ronald and Tim both did their laundry today. Ronald does laundry every 6 days and Tim does laundry every 9 days. How many days will it be until Ronald and Tim both do laundry on the same day again? days. Show Calculator.
Word Problems on L.C.M
LCM Word Problems with Answers. Example 1. Find the lowest number which is exactly divisible by 12, 16. Solution: We have to find the Lcm to find the lowest number exactly divisible by 12,16. LCM= 2 ×2 × 3 × 4=48. Therefore, 48 is the lowest number exactly divisible by 12, 16. Example 2.
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Lowest Common Multiple

Computing the LCM

Image credit: Instagram user brandycandy, Data: Magicicada.org

Least Common Multiple (LCM)

What is the Least Common Multiple (LCM)?

1. Prime Factorization Method

2. Listing Multiples Method

3. Division Method

LCM Formula

Least Common Multiple (LCM) of Two Numbers

Least Common Multiple (LCM) of Three Numbers

Relationship Between LCM and HCF

Least Common Multiple Chart

1. Find LCM of 6 and 8?

2. Find LCM of 14 and 20

3. Find LCM of 18, 24, and 30

AI Generator

Least Common Multiple Simple How-To w/ 9+ Examples!

List Method

Prime Factorization Method

Method Summary

Example — Hard LCM Problem

Example — LCM Of Three Numbers

Worksheet (PDF) — Hands on Practice

Video Tutorial — Full Lesson w/ Detailed Examples

Least Common Multiple (LCM) Worksheets

HCF and LCM Questions

HCF and LCM Questions with Solutions

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Practice Questions

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Word Problems on L.C.M.

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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

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