Assume that a, b, and c are integers and gcd(ab,c)=1. Show that (a,c)=1.


So I started off by writing abx + cy = 1 and ax' + cy' = 1 where x, y, x', and y' are integers. Not sure where to go from here

A divisor of a and c would be a divisor of ab and c. Hence, gcd(ab,c) >= gcd(a,c). If the first is one, then one must be the second.

i think you can do a simple contradiction, right?

suppose gcd(a,c) = x, where x != 1, so we know x > 1

so you know x must divide a

if we multiply a by b, then x must divide a * b. but we know it doesn't since we're given gcd(ab,c) = 1. that's a contradiction.

/edit: looks like i said the same thing as feanur

This post was edited by carteblanche on Feb 22 2014 05:58pm

great thanks for the help