TITLE: GCD : Why does the GCD of two numbers divides their difference?
QUESTION [6 upvotes]: I was working my way through some number theoretic proofs and being a newbie am stuck on this proof :

Why does the gcd of two numbers , say (a,b) - also divides their difference : a-b

My Question :
As I am a novice to induction and proofs , in general , can someone help me out ?

REPLY [4 votes]: For any common divisor, $d$, of $a$ and $b$, there exist integers $x$ and $y$ such that $a=dx$ and $b=dy$.  Then $a-b=dx-dy=d(x-y)$.  S0 $d$ divides $a-b$.