For $d \in \mathbb{Z}$, if $d\mid a$ and $d\mid b$, show that $d\mid(a+b)$ and $d\mid(a-b)$.

Question

Let $d > 0$ and $d \in \mathbb{Z}$. If $d$ divides $a$ and $d$ divides $b$ then I want to show that $d$ divides $a+b$ and $a-b$.

If $d$ divides $a$ then there exist an $m$ such that $a = dm$.

If $d$ divides $b$ then there exists an $n$ such that $b= dn$.

Then $a+b = dm + dn = d(m+n)$, and $a-b = dm - dn = d(m-n)$.

In other words, $a+b$ and $a-b$ are divisible by $d$. Is that it? I start to question myself when its relatively short.

Answer 1

Look here: http://www.cs.usfca.edu/~peter/math300/h02-2key.pdf I hope I could help you ;)