Prove that if $a\mid c$ and $b\mid c$ with $\gcd(a,b)=1$, then $ab\mid c$

Question

I was thinking of using the fact that

$\gcd(a,b) = 1$ implies that $ax + by = 1$ for some integers $x,y$. Then $acx + bcy = c$.

I'm not sure how to proceed.

score 3 · Accepted Answer · answered Jun 07 '15 at 05:08

3

Hint- Write $c=at_1=bt_2$ for some integers $t_1$ and $t_2$

Now do you see it?

answered Jun 07 '15 at 05:08

Bhaskar Vashishth

1

So $a(bt_2)x + b(at_1)y = c$. Hence $ab(t_2x + t_1y) = c$. So $ab|c$ – user226516 Jun 07 '15 at 05:40

1 Answers1