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Explanation:

N = 2⁴ × 3⁵ × 10⁴ = 2⁴ × 3⁵ × 2⁴ × 5⁴ = 2⁸ × 3⁵ × 5⁴

Now, for any number to be a perfect square, it must have even powers.

So, if we consider 2⁸, only powers of 0, 2, 4, 6, 8 can lead to perfect squares i.e. 5 ways

Now, 

If we consider 3⁵, only powers of 0, 2, 4 can lead to perfect squares i.e. 3 ways

If we consider 5⁴, only powers of 0, 2, 4 can lead to perfect squares i.e. 3 ways

So, total number of possibilities = 5 × 3 × 3 ways = 45 ways.

Since we need to find the number of factors greater than 1,

Required number of ways = 45 -1 = 44 ways

Hence, 44.