Let $a, b ∈ Z$ with $gcd(a, b) = 1$, and let $c ∈ Z$. Show the equivalence: $ab | c ⇔ a | c$ and $b | c$.

Question

I'm not sure how to do this question. Please help.

All I know that $gcd(a,b)=1$

But what can I do with this extra information?

"Hint: To prove the implication “$⇐$”, start from $a | c$ and use the fact, due to $gcd(a, b) = 1$, that for any $k ∈ Z$, if $b | ka$ then $b | k$."

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b) Find integers $a, b, c ∈ Z$ with $gcd(a, b)$ does not equal to 1 where the equivalence in ($a$) does not hold.

^ I am so lost from what the question asking

Answer 1

Integers $x, \, y$ exist with $xa+yb=1$, so $k=xka+ybk$ is a multiple of $b$.