Divisors of Product of Coprime Integers

Theorem
Let $a, b, c \in \Z$ be integers.

Let the symbol $\divides$ denote the divisibility relation. Let $a \divides b c$, where $b \perp c$.

Then $\tuple {r, s}$ satisfying:
 * $a = r s$, where $r \divides b$ and $s \divides c$

is unique up to absolute value with:
 * $\size r = \gcd \set {a, b}$
 * $\size s = \gcd \set {a, c}$

Proof
By Divisor of Product, there exists $\tuple {r, s}$ satisfying:
 * $r \divides b$
 * $s \divides c$
 * $r s = a$

We have:
 * $r, s \divides a$

By definition of GCD:
 * $\gcd \set {a, b} \divides r$
 * $\gcd \set {a, c} \divides s$

By Absolute Value of Integer is not less than Divisors:
 * $\gcd \set {a, b} \le \size r$
 * $\gcd \set {a, c} \le \size s$

We also have:

This forces both inequalities to be equalities, i.e.:
 * $\size r = \gcd \set {a, b}$
 * $\size s = \gcd \set {a, c}$

Hence $\tuple {r, s}$ is unique up to absolute value.