Number of perfect square factors greater than 1

First, write $N$ in prime factorized form: $N = 2^4 \times 3^5 \times (2 \times 5)^4 = 2^8 \times 3^5 \times 5^4$

Any factor is of the form $2^a \times 3^b \times 5^c$. For a factor to be a perfect square, $a, b, c$ must all be even integers.

• Possible values for $a \in \{0, 2, 4, 6, 8\}$ (5 choices)
• Possible values for $b \in \{0, 2, 4\}$ (3 choices)
• Possible values for $c \in \{0, 2, 4\}$ (3 choices)

Total perfect square factors $= 5 \times 3 \times 3 = 45$. Excluding 1 (where $a=b=c=0$), the number of perfect square factors greater than 1 is $45 - 1 = 44$.