If $d = \gcd(a, b)$, then $d | a$ and $d | b$

If $d = \gcd(a, b)$, then either $a | b$ or $b | a$

If $d = \gcd(a, b)$, there exist integers $x, y$ with $ax + by = d$

If $d = \gcd(a, b)$, then $\gcd(a/d, b/d) = 1$