Prove the biconditional statement. For $a,b,c$ positive integers, $ac$ divides $bc$ if and only if a divides $b$.
Prove: For a,b,c positive integers, ac divides bc if and only if a divides b
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elementary-number-theory
divisibility
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2Have you tried this yourself? Have you proved one way but not the other? – 2017-02-09
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0Seriously? Just do it. $ac|bc$ means there is some integer $k$ so that $bc = ack$. What does that say about $a$ and $b$. Proofs do not get much easier or more trivial than this. – 2017-02-09
2 Answers
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If $a$ divides $b$, then $b = a \cdot k$ for some integer $k$. Thus if you multiply both sides by $c$, we get the desired $bc = akc$.
On the other hand, if $ac$ divides $bc$, then $bc = akc$ for some integer $k$. Dividing both sides by $c$ yields the desired result.
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The two implications follow directly from definition of division, and the fact that $c$ is positive, so you can divide by it in the two sides of an equallity.