Suppose we know the greatest common divisor, $\gcd(A,B)$, of two numbers $A$ and $B$.
Is there a way that we can find the maximum value of $a^b \bmod N$ where $N$ is any number?
We have a finite range in which both $a$ and $b$ belong so we can just check for all values of $a$ and $b$ that satisfy the given gcd and get the maximum value of $a^b \bmod N$.
But I do feel there might be a better approach to approach this problem.So please guide me.
Also is there a way around this problem if we only know $\gcd(a,b) \bmod N$ for $a$ and $b$ instead of $\gcd(a,b).
Can we still come up with something to determine the maximum value of $a^b \bmod N$.