Divisibility of Sums and Differences

If two integers \(a\) and \(b\) can be divided by a number \(q\), then \(q\) also divides the difference \(a-b\) and the sum \(a+b\).

Proof: As \(q\) divides \(a\) and \(b\) we can also write

\[a=q\cdot x\] \[b=q\cdot y\]

It follows

\[a-b = q\cdot x - q\cdot y = q(x - y)\]

As \(a-b\) is a multiple of \(q\), \(q\) must divide \(a-b\). We can proof it for \(a+b\) in the same way.