The total number of even factors of 25×33×52 is:

Concept

When a whole number is written in prime-factorised form (a product of prime powers), every factor is built by choosing each prime's exponent independently, anywhere from 0 up to that prime's maximum power. So the total number of factors is the product of (each exponent increased by 1). A factor is even only if it contains at least one 2, so for the prime 2 the exponent must be chosen from 1 up to its maximum, which is one fewer choice than usual.

Application

1. Here N = 25 × 33 × 52.

2. For an even factor the power of 2 must be at least 1, so it is chosen from {1, 2, 3, 4, 5}, that is 5 choices.

3. The power of 3 may be 0, 1, 2 or 3, that is 4 choices.

4. The power of 5 may be 0, 1 or 2, that is 3 choices.

5. Since the choices are independent, multiply them: 5 × 4 × 3 = 60.

Cross-check

Counting all factors gives (5+1)(3+1)(2+1) = 72, and the odd factors (power of 2 equal to 0) number (3+1)(2+1) = 12. Subtracting gives 72 − 12 = 60, which agrees with the direct count.