Chapter 4: Factors and Multiples 53There are times when one of the numbers is the GCF of the two numbers. The GCF of 8 and 32
is 8, which is a factor of both 32 and of 8, because every number is a factor of itself. The factors
of 8 are 1, 2, 4, and 8, and the factors of 32 are 1, 2, 4, 8, 16, and 32. 8 is the largest number that
appears on both lists.
At the other extreme, sometimes the only factor the two numbers will have in common is 1.
Remember 1 is a factor of every number. Numbers that have only 1 as a common factor are called
relatively prime. The numbers 12 and 49 are relatively prime because the factors of 12 are 1, 2, 3, 4,
6, and 12, and the factors of 49 are 1, 7, and 49. The only number on both lists is 1.
DEFINITION
Two numbers are relatively prime if the only factor they have in common is 1.For small numbers, making a list of factors isn’t difficult. If the number is small, the list of factors
can’t be too long. But larger numbers can have long factor lists, so it’s important to have a more
efficient method of finding the greatest common factor. That’s when you want to turn to the
prime factorization.
To find the greatest common factor of 128 and 144, first work out the prime factorization of each
number. Factor trees may help you factor to primes. The prime factorization of 128 is seven twos.
128 = 2^7 = 2 v 2 v 2 v 2 v 2 v 2 v 2
The prime factorization of 144 has both twos and threes.
144 = 2^4 v 32 = 2 v 2 v 2 v 2 v 3 v 3
Once you have both prime factorizations, look for any primes that appear in both (these numbers
are underlined). It’s okay to have repeats if the factor repeats in both factorizations. In this case,
there are four 2s that appear in both factorizations. The GCF of 128 and 144 is 2 v 2 v 2 v 2,
which equals 2^4 or 16.
You can find the GCF of 210 and 495 by starting with the prime factorization of each number.
210 = 2 v 3 v 5 v 7
495 = 3 v 3 v 5 v 11
Both factorizations have a 3 and a 5. There’s only one 3 that’s common to both, so only one 3 goes
into the prime factorization. 495 has a second 3, but 210 doesn’t share it. The GCF of 210 and 495
is 3 v 5, or 15.