Suppose n,a,b belong to N. If the gcd of (n,a)=1 and the gcd(n,b)=1, prove the gcd of (n,ab)=1

Question

I would really appreciate any help you can give me. I'm struggling a lot with this problem. Thank you!!

Answer 1

Here is a bit of intuitive reasoning which may help.

 

The fact that (n,a) = 1 and (n,b) = 1 means that n does not have any common factors with a and b, other than 1. This means that n could also not have any common factor with ab other than 1; because if it did, it would be a common factor of a or b.

 

Thus, (n, ab) = 1.