Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$?

Question

Is it true that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$ then $\gcd(ac,b) = 1$?

I know that $\gcd(a,b) = 1$ means that there exist integers $m$ and $n$ such that $am + bn = 1$

Same thing for $\gcd(a,c)$: we have $ax + cy = 1$.

But how can I relate this to $\gcd(ac,b)$?

If you want a specific non-trivial example, let a=3, b=4, and c=10. Note that b and c share 2 as a common factor. — JB King, Sep 28 '14 at 05:41
See also: http://math.stackexchange.com/questions/485013/show-that-gcda-bc-1-if-and-only-if-gcda-b-1-and-gcda-c-1 — Martin Sleziak, Sep 28 '14 at 07:00

score 4 · Accepted Answer · answered Sep 28 '14 at 05:42

4

Let $a=2, b=c=3$. Then $\gcd(a,b)=\gcd(a,c)=1$ and $\gcd(ac,b)=3$.

You can prove that $\gcd(a,bc)=1$, however.

answered Sep 28 '14 at 05:42

Thomas Andrews

1 Answers1