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

Concept.

Any positive integer can be written in prime-factorised form. The total count of its divisors is the product of (exponent + 1) taken over every prime. To count only the ODD divisors, simply discard the factor of 2 entirely (every divisor containing a 2 is even), and apply the same product rule to the remaining odd part.

Application.

1. The number is 25 × 33 × 52. The odd part is everything except the power of 2, i.e. 33 × 52.

2. An odd divisor uses the prime 3 with an exponent from 0 to 3, giving 3 + 1 = 4 choices.

3. It uses the prime 5 with an exponent from 0 to 2, giving 2 + 1 = 3 choices.

4. By the multiplication principle the number of odd divisors is 4 × 3 = 12.

Cross-check.

Total divisors = (5 + 1)(3 + 1)(2 + 1) = 6 × 4 × 3 = 72. Even divisors are those that keep at least one factor of 2, namely 5 × 4 × 3 = 60. So odd divisors = 72 − 60 = 12, which agrees.