Prove $n \mid x + y \implies n \mid x - y$

Question

I would like to prove following proposition

$$n \mid x + y \implies n \mid x - y $$

Attempt:

$$kn = x + y, k \in \mathbb Z $$

Adding $y - y$ to the right side gives

$$kn = x + y - y + y = x - y + 2y$$

Which means

$$n \mid \bigl(x - y\bigr) + 2y$$

Hence $n \mid x - y$. $\Box$

Is it correct?

Answer 1

This is not true.

How do you know $n\mid 2y$? If that would be true than your statment would hold.

Say $5\mid 7+3$ but $5\nmid 7-3$.