7.4 Least Common Multiple (p.170-173) USING PRIME FACTORS

When working to find the LEAST COMMON MULTIPLE (LCM) with large or inconvenient numbers use the prime factors.

1.Find the prime factors for all numbers involved.

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

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

4.Write down that prime number as many times as you counted for it in step 3.

5.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 #2 - 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 #3 - 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 #4 - 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 #5 - 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.