If $\gcd(a, b) = 1$ and $a | bc$, then $a | b$.

For any positive integers $a$ and $b$: $\gcd(a, b) \cdot \text{lcm}(a, b) = a + b$.

If $p$ is prime and $p | ab$, then $p | a$ or $p | b$ (Euclid's Lemma).

$\gcd(a, b) = \gcd(a, b \bmod a)$ for $a > 0$.