algebra help again.. please!!?

Question

prove if gcd(a,c)=gcd(b,c)=1 if and only if gcd(ab, c)=1

Answer 1

It's easiest to think about this in terms of prime power factorisations.

Note: gcd(x, y) = 1 <=> x and y have no common factors (greater than 1) <=> the prime power factorisations of x and y do not have any primes in common.

So, gcd(a, c) = gcd(b, c) = 1 <=> none of the primes in the prime power factorisation of c appear in the factorisations of a or b <=> none of them appear in the product ab <=> gcd(ab, c) = 1.

It is the uniqueness of the prime power factorisation that allows us to equate the statements
"none of the primes in the prime power factorisation of c appear in the factorisations of a or b"
and "none of them appear in the product ab". This is because, given factorisations for a and b, we can write a factorisation for ab using the same primes used in the construction of a and b. Since this is unique and we know the primes in a and b don't divide c, we know there is no prime factor of ab that divides c either.

Answer 2

I can't accept it, and I think you don't want the first "if" there

gcd(a,cb)=1 as well as gcd(ab,c)=1