All Equal

For an array $A$ of length $N$, the integer $K$ is said to be *$A$-good* if the following condition holds: - $\gcd(A_i, K) = \gcd(A_j, K)$ for every $1 \leq i \lt j \leq N$. Define $f(A)$ to be the number of integers between $1$ and $M$ that are *$A$-good.* --- You are given integers $N$ and $M$. Compute the sum of $f(A)$ across all $M^N$ arrays $A$ of length $N$ whose elements lie between $1$