Gats and the Twin Milk Locks

Gats finds $N$ ancient milk chambers labeled $1,2,\ldots,N$. Every chamber has two number locks, and all chambers share a fixed offset $K$.

At the chamber labeled $X$:

• the first lock shows $\gcd(X,N)$;
• the second lock shows $\gcd(X+K,N)$.

A chamber is stable if the product of its two lock values is exactly $N$:

$$\gcd(X,N)\cdot\gcd(X+K,N)=N.$$

Count the stable chamber labels $X$ satisfying $1 \le X \le N$.

10 1

4