# Possible remainders of a square

**Number Theory**Level 5

Let \[N= 3 \times 5 \times 7 \times 11 \times 13\] There are \(X\) possibilities for the units digit of a perfect square when it is represented in base \(N\). Compute the last three digits of \(X\) (in base 10).

**Details and assumptions**

We are talking about \(3, 5, 7, 11, 13\) in their decimal representations.

You might use the fact that \(3, 5, 7, 11, 13\) are all primes.