Prove or disprove if $a\mid b+c$, then either $a\mid b$ or $a\mid c$

Question

Prove the following statement, otherwise give a counter example If $a\mid b+c$, then either $a\mid b$ or $a\mid c.$

I could disprove this using a counter example. Let's say $b=67,c=1,a=2$.

$2$ divides $67+1=68$. However, $2$ does not divide $67$ or $1$.

Is this a good counter example? Is there a better way of disproving this statement?

Answer 1

What is a good counterexample, and what is a bad one? It should be correct, this is the most important part. Yes, $2\mid 67+1$, but not $2\mid 1$ or $2\mid 67$. Of course, one can take an "easier" counterexample, e.g., $$ 2\mid 3+3 $$ but $2\nmid 3$.

Answer 2

Your example is fine. $\text{ }$

Answer 3

Alternatively, consider $b=a+1$ and $c=a-1$.