Corollary 1.3.7.

Corollary 1.3.7. Let a, b 1 , and b 2 be integers such that a and b 1 are coprime, and so are a and b 2 . Then a is coprime with b 1 b 2 . Proof. The following relations hold: 1 = αa + β 1 b 1 , 1 = ′ α a + β2 b2 . By multiplying them, we get 1 = (αα ′ a + α β 2 b 2 + α ′ β 1 b 1 )a + ( β 1 β 2 )(b 1 b 2 ), proving the claim. ⊔⊓ Corollary 1.3.8. Let a, b, and n be integers such that a | n, b | n and GCD(a, b) = 1. Then ab | n. Proof. We have n = n 1 a = n 2 b. Moreover, a r...