Greatest Common Divisor

Definition (GCD)

The greatest common divisor of two integers $b,c$ is denoted as $\gcd (b,c)$, the largest integer which divides both $b$ and $c$.

Lemma (Remainder by GCD)

In general,

$$\gcd (b,bq+r)=\gcd (b,r)$$

for any integers $q,r$. This observation leads us the Euclidean Algorithm.

Example 1:

$$\gcd (0,a)$$

Since every number divides $0$, then this $\gcd$ of just $a$ is $a$.