How many positive integers $n$ satisfy $127 \equiv 7 \pmod{n}$? $n=1$ is allowed.

Naturally, any integer larger than 127 will return , and of course , so we restrict the possible solutions to .

Now,



is the same as saying there exists some integer such that



We have



which means that any that satisfies the modular equivalence must be a divisor of 120, of which there are 16: .

In the cases where the modulus is smaller than the remainder 7, we can see that the equivalence still holds. For instance,



(If we're allowing , then I see no reason we shouldn't also allow 2, 3, 4, 5, 6.)