# 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 want to use the fact that $$3, 5, 7, 11, 13$$ are all primes.
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