A 4-digit number N has exactly 15 distinct divisors. What is the total number of distinct divisors of N^2?

Explanation

If N has 15 divisors, 15 = 5 \times 3, so N must be of the form p^4 q^2 (as p^{14} is too large to be a 4-digit number). Squaring gives N^2 = p^8 q^4. The number of divisors of N^2 is (8+1)(4+1) = 9 \times 5 = 45.

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