Theorem 8.4: If a and b are relatively prime and a divides bc, then a divides c.

Proof: If a and b are relatively prime, then by Definition 8.3, their greatest common factor is 1. By Theorem 8.2 we can then write 1 as a sum or difference of multiples of a and b. When we multiply c by this representation of 1 we get c as a sum or difference of multiples of ac and bc. a divides ac by Definition 8.1 and it divides bc by hypothesis so it divides any sum or difference of multiples of ac and bc, by Theorem 6.10 and Theorem 7.5, including c itself.