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Least Common Multiple?

The least common multiple (LCM) is a concept that describes a relationship between __ 2 or more__ numbers.

The multiples of a number are all of the numbers that you get if you multiply or skip-count by that number.

For example...

- The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18,...
- The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48,...

So, every number has an infinite number of multiples.

The least common multiple of a pair of numbers is the smallest number that is a multiple of ** both** of the numbers in the pair.

In the example above, the LCM of 2 and 6 would be 6 because that is the smallest multiple that they have in common. They also have other multiples in common (like 12 and 18) but 6 is the smallest.

There are several different ways to find the least common multiple.

- List Method
- Prime Factorization Method
- GCF Method

Most people think that the list method is the easiest way to find the LCM and I agree that it is the easiest method when you are working with small numbers.

However, prime factorization is easier when you are trying to find the LCM of large numbers or when you are trying to find the LCM of 3 or more numbers.

I also really like the prime factorization method because it reveals very interesting relationships between the numbers. And these relationships make it really easy to find the least common denominator of fractions and rational functions.

I don't use the GCF method very often, but it can be useful if you already know the greatest common factor of the numbers you are working with. This method only works if you are finding the LCM of 2 numbers.

with the List Method

In most cases, the list method is the easiest way to find the LCM. It is especially easy if you are working with small numbers.

To find the least common multiple with the list method...

- Start a list of multiples for each number.
- Continue each list until you find a multiple that the numbers have in common.

Step 1: Start a list of multiples for each number.

Multiples of 8: 8, 16, 24, 32,...

Multiples of 28: 28, 56, 84,...

Step 2: Continue each list until you find a multiple that the numbers have in common.

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72,...

Multiples of 28: 28, 56, 84,...

Answer: 56 is the first multiple that 8 and 28 share, so the LCM is 56.

with the Prime Factorization Method

The prime factorization method is easier than the list method when you are finding the LCM of large numbers. It is also really helpful if you need to find the LCM of 3 or more numbers.

To find the least common multiple with the prime factorization method...

- Find the prime factors of each number.
- List each prime factor the
number of times it occurs in either number.**maximum** - Multiply these prime factors together.

Step 1: Find the prime factors of each number.

\(12=2\times2\times3\)

\(40=2\times2\times2\times5\)

Step 2: List each prime factor the __ maximum__ number of times it occurs in either number.

The prime factor 2 occurs twice in 12 and thrice in 40.

Thrice is the maximum number of times 2 occurs.

So, we need three 2's in our LCM.

The prime factor 3 occurs once in 12 and never in 40.

Once is the maximum number of times 3 occurs.

So, we need one 3 in our LCM.

The prime factor 5 occurs never in 12 and once in 40.

Once is the maximum number of times 5 occurs.

So, we need one 5 in our LCM.

Step 3: Multiply these prime factors together.

LCM of 12 & 40 \(=2\times2\times2\times3\times5=120\)

Answer: 120 is the least common multiple of 12 and 40.

I know the second step of finding the LCM with the prime factorization method can be a little confusing.

To make it easier, I like to think of the prime factors as legos. Each of the original numbers is a lego creation and the LCM is the lego set that would allow you to build both lego creations individually.

So, if you used this analogy for the example above, the number 12 "lego creation" requires two 2-legos and one 3-lego. The number 40 "lego creation" requires three 2-legos and one 5-lego.

A lego set with three 2-legos, one 3-lego, and one 5-lego would allow you to build both lego creations individually, though not necessarily at the same time. That lego set represents the LCM of 12 and 40.

with the GCF Method

To find the least common multiple with the GCF method...

- Find the GCF of the numbers.
- Find the product of the numbers.
- Divide the product by the GCF.

Note: This method only works if you are finding the LCM of two numbers.

Step 1: Find the GCF of 48 & 30.

\(48=\color{black}{2}\times2\times2\times2\times\color{black}{3}\)

\(30=\color{black}{2}\times\color{black}{3}\times5\)

GCF of 48 & 30 \(=\color{black}{2\times3}=6\)

Step 2: Find the product of 48 & 30.

\(48\times30=1440\)

Step 3: Divide the product by the GCF.

\(1440\div6=240\)

Answer: The LCM of 48 & 30 is 240.

of Three or More Numbers

Usually, you will only be asked to find the least common multiple of a pair of numbers.

However, if you are ever asked to find the LCM of three or more numbers, you can use the list method or the prime factorization method and the process will be basically the same.

You just have to make sure that __ ALL__ of the numbers have the multiple in common.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64,...

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80...

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90,...

\(4=2\times2\)

\(5=5\)

\(6=2\times3\)

\(LCM=2\times2\times3\times5=60\)

Answer: Regardless of which method you use, 60 is the least common multiple of 4, 5, and 6.

The least common multiple very useful when you are adding or subtracting fractions and have to find common denominators.

It is also very helpful when you are adding or subtracting rational functions and have to find common denominators with variables.

The least common multiple can be used to describe the overlap of cycles in real life situations.

For example, let's say you wash your dishes every other day, you wash your sheets every week, and you wash your floor every 5 days. The LCM of 2, 7, and 5 would tell you how often you will wash your dishes, sheets, and floor all on the same day.

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