Prove that if $c\mid a-b$ and $c\mid a+b$ then $c \mid a$

Question

Let $a, b, c$ be integers.

$c\mid a-b$ implies $a-b = cm$, $m \in \mathbb{Z}$

$c \mid a+b$ implies $a+b = cn$, $n \in \mathbb{Z}$

$(a-b)+(a+b) = cm + cn = c (m+n)$

$2a = c(m+n)$

Then I have no idea.

Answer 1

In point of fact, this is false: $2\mid 3-1$, $2\mid 3+1$ and $2\nmid 3$.