# Prove that $p\mid b$ or $p\mid c$

If $a,b,c$ are integers and $p$ is a prime that divides both $a$ and $a +bc$, prove that $p\mid b$ or $p\mid c$.

The way I'm trying to think of it, and I might be completely wrong, is using a theorem that says:

Let $p$ be an integer with $p\neq 0,\pm 1$. Then $p$ is prime if and only if $p$ has this property: Whenever $p\mid bc$, then $p\mid b$ or $p\mid c$

• Should that not be 'prove that p|b or p|c'? – lemon Aug 24 '14 at 21:03
• The word "or" in your title is correct. The word "and" in the gray box is incorrect. – Will Jagy Aug 24 '14 at 21:03

Since $p|a$ then there exists a $k$ such that $a=kp$. Similarly, there exists a $k'$ such that $a+bc=k'p$. It follows that $bc=p(k'-k)$ and so $p|bc$. Then you apply the theorem you state in your question...