If $c | ab$, then $c | a$ or$ c | b$

Question

I need help proving/disproving the implication, If $c | ab$, then $c | a$ or $c | b$

So far, I got Assume $c | ab$ then $ab= cl$ for some integer $l$ Now what should my next step be?

Answer 1

$4|20 =2\times 10$ but 4 doesnot divide 2 and 10

Answer 2

As suggested in the comments the statement is only true when $c$ Is prime.

I will use $p$ instead of $c$ so that it will be easier to remember that it is prime.

Go for the contra positive of the statement. So assume that $p$ (prime) does not divide both $a$ and $b$. By division algorithm, $$ a=pq + r \qquad b=pt+s.$$ Where $r$ And $s$ are non-zero remainders that are less than $p$.

Then $$ ab = p(pqt + qs + rt) + rs .$$

Now try to show that $ab$ will have a non-zero remainder when divided by $p$. This is where the fact that $p$ is prime will be used.