Possible remainders of a square

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.

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