Suppose a is a number > 1 with the following property: for all b,c, if a divides bc and a does not divide b, then a divides c. Show a must be prime

Question

I know that $ax+by = d$ by Bezout's theorem but I really don't know how to proceed with this one. I tried saying $bc = ak_1$ and $c = ak_2$

Answer 1

Suppose that $a$ is not prime. Let $d$ be the smallest prime divisor of $a$.

Taking $b=d$ and $c = \frac{a}{d}$, one then has $a$ divides $bc$ and $a$ doesn't divide $b$, but $a$ does not divide $c$.