Algebra: Proving (a^2,b^2)=1 Using GCD Proof

[maphec]

(a,b)=d means d is the GCD of a and b

Question:

Let (a,b)=1

Prove: (a^2,b^2)=1



The hint that we were given is to prove this by contradiction ... but, I have no idea how to go about even starting this proof ... Any and all help would be greatly appreciated!

 

[SammyS]

Suppose that (a,b) = 1 and that (a²,b²) = k, where k > 1 ...

 

[maphec]

Proof:

Supposed n is prime and that n|ab
therefore, n|a or n|b, but not both

Case 1: n|a and n does not divide b
therefore n|a^2
since n does not divide b, n does not divide b^2

Case 2: n|b and n does not divide a
therefore n|b^2
since n does not divide a, n does not divide a^2

Thus, a^2 and b^2 have no common divisors
and therefore gcd(a^2,b^2)=1