Least common multiple

This lesson will teach you 3 methods for finding the least common multiple (LCM) of two whole numbers. We will start with a definition of the word multiple.

The multiples of a number are the answers that you get when you multiply that number by the whole numbers except 0.

Remember that the whole numbers are all numbers from 0 to infinity.

Whole number = {0, 1, 2, 3, 4, 5, 6, 7, 8, …}

So multiply any number by 1, 2, 3, 4, 5, 6, 7, 8, … to get the multiples of that number.

For instance, you get the multiples of 4 by multiplying 4 by 1, 2, 3, 4, 5, …

I put the dots to show that the sets of whole numbers continues forever.

The answer is { 4, 8, 12, 16, 20, …}

In the same way, the multiples of 9 are all the numbers that you get when you multiply 9 by 1, 2, 3, 4, 5, 6, …

After you do that, you will get { 9, 18, 27, 36, 45, 54, …}

The LCM of two numbers is the smallest number that is a multiple for both numbers.

Method #1: Set intersection method

Example: Find LCM of 6 and 9.

First list all the multiples of 6.

You get {6, 12, 18, 24, 30, 36, 42, 48, 54, 60,….}

Next, list all the multiples of 9.

You get {9, 18, 27, 36, 45, 54, 63, 72, 81, 90….}

Pull out all the common multiples or find the intersection of the two sets.

The common multiples are {18, 36, 54, …} Looking at the list of common multiples immediately above, you can see that the smallest number that is a multiple of both 6 and 9 is 18.

Of course, 36 is also a common multiple of 6 and 9. However, it is not the smallest common multiple.

Example: Find LCM of 2 and 3.

Multiples of 2 are {2, 4, 6, 8, 10, …}

Multiples of 3 are {3, 6, 9, 12, 15, …}

The least common multiple of 2 and 3 is 6.

You can also write LCM(2,3) = 6 .

Method #2: My teacher’s method

Find LCM( 6, 9) and LCM (120, 180)

Technique:

Start by dividing each number by 2. (If 2 does not work, start with 3 instead, and so forth …)

Keep dividing by 2 until 2 does not work anymore.

When 2 does not work anymore, divide by 3.

When 3 does not work anymore, divide by 4.

Keep doing this until you can no longer divide.

LCM = The product of all the numbers on the left of the red line.

Method #3: Prime factorization method

Find LCM(120, 180)

First, find the prime factorization of the numbers.

120 = 12 × 10 = 2 × 2 × 3 × 2 × 5 = 2³ × 3¹ × 5¹

180 = 18 × 10 = 2 × 3 × 3 × 2 × 5 = 2² × 3² × 5¹

The least common multiple will be 2^x × 3^y × 5^z

x is the bigger exponent of 2³ and 2²

y is the bigger exponent of 3¹ and 3²

y is the bigger exponent of 5¹ and 5¹

The least common multiple is 2³ × 3² × 5 = 2 × 2 × 2 × 3 × 3 × 5 = 360