When does $a\mid c$ and $b\mid c$ imply $ab\mid c$?

Question

For example, $$2\mid12,3\mid12,6\mid12$$ But,
$$4\mid12,6\mid12,24 \nmid 12$$ When does it work?

Answer 1

The proposition you are searching for is valid if $gcd(a,b)=1.$
So it goes like this:
$a|c$ and $b|c$ and $gcd(a,b)=1 \Rightarrow ab/c$

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Hence, $a$ and $b$ should be relatively prime to achieve the needed result. The counterexample you mentioned did not satisfy the proposition because gcd(6,4)=2.
Note: If you are interested in proving it, go ahead and try using the Bezout Identity!