Common Divisor Divides Difference/Proof 1

Theorem

Let $c$ be a common divisor of two integers $a$ and $b$.

That is:

$a, b, c \in \Z: c \divides a \land c \divides b$

Then:

$c \divides \paren {a - b}$

Proof

Let $c \divides a \land c \divides b$.

From Common Divisor Divides Integer Combination:

$\forall p, q \in \Z: c \divides \paren {p a + q b}$

Putting $p = 1$ and $q = -1$:

$c \divides \paren {a - b}$

$\blacksquare$