Algebra, gcd help?

Question

Prove that if a,b are in Z, but a and b are different from 0, then gcd(a/gcd(a,b), b/gcd(a,b))=1.
I tried, but couldn't do it. Please help me.

Answer 1

The gcd of two numbers is the product of the prime numbers they have in common in their prime factorization.

a' = a / gcd ( a, b ) is a number that has no prime factors in common with b. In other words, you have a number that once had prime factors in common with b, but they have now been divided out.

Similarly, b' = b / gcd ( a, b ) is a number that has no prime factors in common with a.

Now, do a' and b' have common factors? Let's say that they do and that the common factor is q. But if a' and b' have factor q in common, then a and b must also have factor q in common, and that factor would have been divided out by gcd ( a, b ).

In other words, a' and b' are now "relatively prime" and their gcd is one.