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

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