Possible remainders of a square

Let

N=3×5×7×11×13.N= 3 \times 5 \times 7 \times 11 \times 13.

There are XX possibilities for the units digit of a perfect square when it is represented in base N.N. Compute the last three digits of XX (in base 10).

Details and Assumptions:

  • We are talking about 3,5,7,11,133, 5, 7, 11, 13 in their decimal representations.
  • You might want to use the fact that 3,5,7,11,133, 5, 7, 11, 13 are all primes.
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