For all integers a, b, and c, if ab | c then a | c and b | c

Question

Prove or disprove the following:

For all integers a, b, and c, if ab | c then a | c and b | c

I'm having trouble proving the above. It seems to be obviously true in my head (but only because of all the examples I can think of), but I'm having trouble proving it.

Answer 1

Hint. By definition, $x\mid y$ if and only if there exists an integer $z$ so that $y=xz$. So if $ab\mid c$, write $c=(ab)(d)$ for some integer $d$. What does this tell you when you observe that $(ab)(d)=a(bd)=b(ad)$?

Answer 2

$a|ab, ab|c \therefore a|c$
$b|ab, ab|c \therefore b|c$

Q.E.D.

Note that the converse ($a|c, b|c \implies ab|c$) isn't true, let $a=b=c\ne\pm1$ for example.