Ms. Spooner

Least Common Multiple

LCM

Standard

6.N.8 Apply number theory concepts–including prime and composite numbers, prime factorization, greatest common factor, least common multiple, and divisibility rules for 2, 3, 4, 5, 6, 9, and 10–to the solution of problems

Cluster

C2

Vocabulary

Multiple- a mulitple of a number is the product of the number and any non zero number

Common Multiple- a multiple shared by two or more numbers

Least Common Multiple- the smallest of the common multiples

Objective

To find least common multiples

Lesson

The least common multiple of two or more non-zero whole numbers is actually the smallest whole number that is divisible by each of the numbers.
There are two widely used methods.

Method 1

Simply list the multiples of each number (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.


Example: Find the least common multiple for 5, 6, and 15.
First we list the multiples of each number.

Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,...

Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,...

Multiples of 15 are 30, 45, 60, 75, 90,....

Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
Therefore, the least common multiple of 5, 6 and 15 is 30.

Method 2

To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following...

Count the number of times each prime number appears in each of the factorizations.

For each prime number, take the largest of these counts.

Write down that prime number as many times as you counted for it in step 2.

The least common multiple is the product of all the prime numbers written down.

Example: Find the least common multiple of 5, 6 and 15.

Factor into primes

Prime factorization of 5 is 5

Prime factorization of 6 is 2 x 3

Prime factorization of 15 is 3 x 5

Notice that the different primes are 2, 3 and 5.

Now, we do Step #1 - Count the number of times each prime number appears in each of the factorizations...

The count of primes in 5 is one 5

The count of primes in 6 is one 2 and one 3

The count of primes in 15 is one 3 and one 5

Step #2 - For each prime number, take the largest of these counts. So we have...

The largest count of 2s is one

The largest count of 3s is one

The largest count of 5s is one

Step #3 - Since we now know the count of each prime number, you simply - write down that prime number as many times as you counted for it in step 2.

Here they are...

2, 3, 5

Step #4 - The least common multiple is the product of all the prime numbers written down.

2 x 3 x 5 = 30

Therefore, the least common multiple of 5, 6 and 15 is 30.

Practice 

If you need more practice, use the online tutorial in our Text Book.