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.