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.
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.
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.
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...
Step 1: Start a list of multiples for each number.
Step 2: Continue each list until you find a multiple that the numbers have in common.
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...
Step 1: Find the prime factors of each number.
Step 2: List each prime factor the maximum number of times it occurs in either number.
Step 3: Multiply these prime factors together.
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.
To find the least common multiple with the GCF method...
Note: This method only works if you are finding the LCM of two numbers.
Step 1: Find the GCF of 48 & 30.
Step 2: Find the product of 48 & 30.
Step 3: Divide the product by the GCF.
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.
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.