Philippine Mathematical Olympiad, National Orals

Solution:
Note that $N = 2^{3} \cdot 5 \cdot 101 \cdot 1009$ and so $N$ has $4 \cdot 2 \cdot 2 \cdot 2 = 32$ factors. On the other hand, $N^{2} = 2^{6} \cdot 5^{2} \cdot 101^{2} \cdot 1009^{2}$ and so $N^{2}$ has $7 \cdot 3 \cdot 3 \cdot 3 = 189$ factors. Hence, $189 - 32 = 157$ of these factors of $N^{2}$ do not divide $N$.