How many factors of 2^5 3^6 5^2 are perfect squares?

Step-by-step solution

The given number is N = 2⁵ 3⁶ 5² . A factor of N is of the form 2^ a 3^ b 5^ c , where 0 a 5 , 0 b 6 , and 0 c 2 . For the factor to be a perfect square, the exponents a, b, c must be even integers. The possible values for a are 0, 2, 4 (3 values). The possible values for b are 0, 2, 4, 6 (4 values). The possible values for c are 0, 2 (2 values). The total number of such factors is the product of the number of choices for each exponent: 3 4 2 = 24 . Answer: 24