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# Least Common Multiple

LCM stands for “Least Common Multiple.” Finding the smallest multiple evenly divisible by two or more integers is a mathematical concept. In other words, the LCM of a set of numbers is the smallest positive integer that can be divided by each of those numbers without leaving a remainder.

List of Contents:

• What is a multiple?
• What is the lowest common multiple?
• Prime Factorization Method
• Listing Multiples
• Division Method
• FAQs

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## What is a multiple?

A multiple of a number is a number that can be obtained by multiplying that number by a whole number.
For example, multiples of 6 are 6, 12, 18, 24, 30…etc.

## What is the lowest common multiple?

The lowest common multiple (LCM) is the smallest multiple that two or more numbers have in common. In other words, it’s the smallest positive number divisible by both (or all) of the given numbers without leaving a remainder.

For example, let’s find the LCM of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, …
Multiples of 6: 6, 12, 18, 24, …
The smallest multiple in both lists is 12, so 12 is the LCM of 4 and 6.

Finding the lowest common multiple:

To find the LCM of two or more numbers, we can use these given methods:

• Prime factorization method
• Listing multiples
• Division Method

## Prime Factorization Method

The prime factorization method is a mathematical technique used to express a positive integer as a product of its prime factors. In other words, it involves breaking down a number into the smallest prime numbers that, when multiplied together, equal the original number.

Let’s say we want to find the LCM of 12 and 18.

Step 1: Prime Factorization
Prime factorize each number:
12 = 2 * 2 * 3
18 = 2 * 3 * 3

Step 2: List Prime Factors with Highest Powers
List all the prime factors with their highest powers from both numbers:
22 (since it appears with the highest power of 2 in 12)
32 (since it appears with the highest power of 2 in 18)

Step 3: Multiply the Prime Factors
Multiply these prime factors together:
LCM = 22 X 32 = 4 X 9 = 36

Hence, the LCM of 12 and 18 is 36.

## Listing Multiples

Finding the lowest common multiple (LCM) by listing multiples involves identifying the multiples of each number and then finding the smallest multiple that is common to both (or all) of the given numbers.

Here’s how you can do it step by step:

Let’s find the LCM of 8 and 12

Step 1: List Multiples
List the multiples of each number until you find a common multiple:
Multiples of 8: 8, 16, 24, 32, 40, …
Multiples of 12: 12, 24, 36, 48, 60, …

Step 2: Identify the Smallest Common Multiple
Identify the smallest multiple that appears in both lists:
24 is the smallest multiple that appears in both the lists.

Step 3: LCM
The smallest common multiple is the LCM of the two numbers
LCM = 24

So, the LCM of 8 and 12 is 24.

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## Division Method

The division method to find the Least Common Multiple (LCM) of two or more numbers involves finding the LCM by repeatedly dividing the numbers by their greatest common divisor (GCD).

Here are the steps to find the LCM of two numbers using the division method:

Step 1: Find a prime number which is a factor of at least one of the given numbers. Write this prime number on the left of the given numbers.

Step 2: If the prime number in step 1 is a factor of the number, then divide it by the prime and write the quotient below it. If the prime number in step 1 is not a factor of the number, then write the number in the row below as it is. Keep on dividing until 1 is left in the last row.

Step 3: Then we multiply the numbers we get on the left. The product of all these prime numbers is the LCM.

## Least Common Multiple FAQS

#### What is the Lowest Common Multiple (LCM)?

The LCM of two or more numbers is the smallest multiple all those numbers have in common.

#### Why is the LCM important in mathematics?

The LCM is important for various mathematical operations, like adding and subtracting fractions with different denominators, solving equations, and working with ratios.

#### How can I find the LCM of two numbers?

You can find the LCM using methods like listing multiples, prime factorization,or division method.

#### Why is finding the LCM important when adding fractions?

You need to find a common denominator when adding fractions with different denominators. The LCM provides the smallest common multiple that can be used as the least common denominator.

#### Can the LCM of two numbers be smaller than either of the numbers?

No, the LCM always equals or exceeds the considered numbers.

#### How can I check if I've calculated the LCM correctly?

You can verify your LCM by ensuring it is divisible by each original number and that it is the smallest multiple they have in common.

#### Is the LCM the same as the greatest common divisor (GCD)?

The LCM is about finding common multiples, while the GCD is about finding the largest common factor/divisor between numbers.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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