So I'm given the following as a problem:
When $p$ and $q$ are distinct odd primes and $N = pq$, the points in $Z^∗_N$ have either zero or four square roots. A quarter of the points have four square roots; the rest have no square root. The four square roots of $x ∈ Z^∗_N$ will look like $±a, ±b$. (Of course, −a means N − a since we’re working modulo N.) Suppose that I give you an efficient deterministic algorithm S that, on input x that has square roots, finds some square root. (If x does not have a square root, it returns ⊥.) Use S to make an efficient probabilistic algorithm F that factors N.
I am not asking for a solution to the problem, rather an understanding of what "four square roots" could possibly mean? How can a single element $x∈ Z^∗_N$ have "four square roots"?