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.