Let us have two numbers, a, b, so that gcd(a,b)=d.
Let us have two prime numbers k,l so that gcd(ak, bl)=d.
Is this valid for all prime numbers or is there any conditions that must be met?
I guess that it is valid for all prime numbers except those that are divisors of a and b, i.e. kcannot be the prime number which is also divisor of b and lcannot be the prime number which is also divisor of a, but I do not know how to prove it.
Could anyone help? Thank you.