The number of factors of 35×58×72 that are perfect squares is ____________.

Question

Accepted Answer

A factor is a perfect square iff each prime's exponent is even.

For $3^{5}$ : even exponents are $0, 2, 4$ → 3 choices.

For $5^{8}$ : $0, 2, 4, 6, 8$ → 5 choices.

For $7^{2}$ : $0, 2$ → 2 choices.

Total perfect-square factors = $3 \times 5 \times 2 = 30$ .