## Ring of Integers Isomorphism Problem

The problem statement, all variables and given/known data
Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB.

The attempt at a solution
I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to conclude, since ZN and ZA x ZB have the same cardinality and f is injective, that f is surjetive? In any case, given an (a, b) in ZA x ZB, I've been trying to find an n such that f(n) = (a, b) without success. Any tips?
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 Good tip. Thanks.

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