GCD of Integer with Integer + n

Theorem
Let $a \in \Z$ be an integer.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Then:
 * $\gcd \set {a, a + n} \divides n$

where:
 * $\gcd$ denotes the greatest common divisor
 * $\divides$ denotes divisibility.

Proof
Let $g = \gcd \set {a, a + n}$.

By definition of $\gcd$, there exist $b, b' \in \Z$ such that:


 * $a = g b$
 * $a + n = g b'$

Therefore:

Since $b' - b \in \Z$, it follows by definition of divisibility that:


 * $g \divides n$

as desired.