Abstract Algebra: Theory and Applications · Problem 2.4.27

Groups — Thomas Judson

Mathematics — Abstract Algebra · euclid-lemma-generalization

Problem

Let

a, b, c \in \mathbb{Z}

. Prove that if

\gcd(a,b) = 1

and

a \mid bc

, then

a \mid c

. Hint. Since

\gcd(a,b) = 1

, there exist integers

r

and

s

such that

ar + bs = 1

. Thus,

acr + bcs = c