# Help understanding divisibility proofs

Can someone please explain how to do this problem?

Prove or disprove the statement “If $a \mid b$ and $c \mid d$, then $(a + c)\mid(b + d)$”.

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Experimentation with such things can be very important, both in a search for a counterexample and a search for a proof. In your case, in seconds you will find a counerexample. – André Nicolas Oct 3 '12 at 16:01

Hint $\$ By the mediant inequality $\rm\:\frac{b}a < \frac{d}c\:\Rightarrow\:\frac{b}{a} < \frac{b+d}{a+c} < \frac{d}{c}.\:$ Therefore, when the extreme terms are consecutive integers, the middle term is not an integer.

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Take $a=2,b=4,c=3,d=3$, then what can you say ?

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This is funny ^^ – mick Oct 3 '12 at 16:02
Yes, counter examples are easy to find, but how do I write the proof for one that is true? I can't possibly test every example. Like this one: Prove or disprove the statement “If a | c and b | d, then ab | cd”. – Julie Oct 3 '12 at 19:12
Assume $a \mid c$ and $b \mid d$. Then there exist integers $l,m$ with $am = c$ and $bl = d$. Find integer $q$ with $(ab)*q=cd$ – Stefan Smith Oct 3 '12 at 22:38