How to prove divisibility of the difference between two numbers.

Question

Recently I have come across a statement saying that if $x$ and $y$ are divisible by $a$, then $x - y$ is also divisible by $a$.

How can I prove this? Does it also apply to sum of $x$ and $y$ ?

Answer 1

If $x=aq$ and $y=ap$ then $x-y=aq-ap=a(q-p)$ by the distributive law.

Answer 2

So, we have integers $X,Y$ such that $$\frac xX=\frac yY=a$$

$$\implies x-y=a(X-Y)\iff\frac{x-y}a=X-Y$$ which is an integer

In fact we can prove $$cx-dy$$ is also divisible by $a$ for integers $c,d$

Here $c=d=1$