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Let $a$,$b\in \mathbb{Z}$, not both zero. Let $S = \{n\in\mathbb{Z}|n=ax+by\text{ for some }x,y\in\mathbb{Z}\}$ Let $d=(a,b)$. Prove that $S$ is the set of all integer multiples of d.

Alex
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  • @Alex I see you're new to the site. Usually, you want to provide some context when you ask questions on here as well as what you've tried so far. –  Apr 04 '17 at 10:03

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This is well known basic algebra problem. And since there's already been similar posts about this problem, I ll give you a simple, but very useful hint.

Hint: Try to use Euclid's algorithm