How do you use prime factorization to find the gcf of two numbers?

Answer 1

Write the numbers in their prime factorisations. the GCF is the product of the common primes (each time they appear).

Example: 30 and 42

2 x 3 x 5 = 30

2 x 3 x 7 = 42

Select the common factors.

2 x 3 = 6, the GCF

Example: 24 and 60

24 = 2 × 2 × 2 × 3

60 = 2 × 2 × 3 × 5

GCF = 2 × 2 × 3 = 12

The prime 2 appears twice in both numbers.

If you use the power format of the prime factorisation, the GCF is the product of each of the common primes to the LOWEST power across the numbers.

For 24 and 60 this gives:

24 = 2³ × 3

60 = 2² × 3 × 5

→ GCF = 2² × 3 = 12

As the lowest power of 2 across the numbers is 2²,

the lowest power of 3 across the numbers is 3 = 3¹,

5 is not common, but for completeness, 24 = 2³ × 3¹ × 5⁰ since 5⁰ = 1, so the lowest power of the prime 5 across the numbers is 5⁰:

→ GCF = 2² × 3¹ × 5⁰ = 4 × 3 × 1 = 12.