If I have two co-prime integers. $a, b$. Suppose that the product of these two integers is $c$.
Further suppose that I have a further product of two co-primes, so $d = af$. Now if I multiply these together to obtain:
$$cd = a^2bf.$$
From this i then say that $\gcd(cd, a^4) = a^2$.
Its clear that $a^2|a^4$ and $a^2|cd$.
Why can't the $\gcd = a^3$?
What I can't get my head around is that I know $a, b$ and $a, f$ are coprime (from this I can't deduce that the three integers are mutually coprime, can I? ), but surely there must exist a product $bf$ which has factor $a$?
My idea is clearly flawed otherwise $a^3$ would be the answer. I really would like to put this idea to rest.