I'm trying to find a proof for the following result.
Consider a sum $a+b=c$. If $p$ divides $c$ then either
a) both $a$ and $b$ are divisible by $p$
b) both $a$ and $b$ are not divisible by $p$
Another case that seems to be a occurrence of the same rule (?) is
two things that are congruent modulo $c$ are either both divisible by $c$ or both not
I wasn't able to find any proof nor a lemma that would incorporate this rule.