GCD of Integer and Divisor

Theorem
Let $$a, b \in \mathbb{Z}^*_+$$, i.e. integers such that $$: a, b > 0$$.

Then $$a \backslash b \Longrightarrow \gcd \left\{{a, b}\right\} = a$$.

Proof

 * $$a \backslash b$$ by hypothesis, $$a \backslash a$$ from Integer Divisor Results.

Thus $$a$$ is a common divisor of $$a$$ and $$b$$.


 * Note that $$\forall x \in \mathbb{Z}: x \backslash a \Longrightarrow x \le \left|{a}\right|$$ from Integer Absolute Value Greater than Divisors.

As $$a$$ and $$b$$ are both positive, the result follows.