## Proof That Two Numbers Divided by Their Greatest Common Divisor are Relatively Prime

Theory
If
$a, \; b$
are integers with greatest common divisor
$d$
then the greatest common divisor of
$\frac{a}{d}, \; \frac{b}{d}$
is 1.
Proof
If the greatest common divisor of
$a, \; b$
is d then we can find integers
$m, \; n$
such that
$am+bn=d$
. Dividing by
$d$
gives
$\frac{a}{d}m+ \frac{b}{d}n=1$
. Then by this theortem
$\frac{a}{d}, \; \frac{b}{d}$
have greatest common divisor 1 and are relatively prime.