The total number of factors of 25 × 33 × 52 excluding one and itself is:

Concept

For any number written in prime-factorised form ap × bq × cr, the total count of positive divisors (factors) equals the product of each exponent increased by one: (p + 1)(q + 1)(r + 1). Every divisor is built by choosing an exponent for each prime independently, from 0 up to its maximum, which gives that many choices per prime.

Application

Here the number is 25 × 33 × 52, with exponents 5, 3 and 2.

1. Increase each exponent by one: 5 + 1 = 6, 3 + 1 = 4, 2 + 1 = 3.

2. Multiply these counts to get the total number of factors: 6 × 4 × 3 = 72.

3. The phrase “excluding one and itself” removes exactly two of those factors — the divisor 1 and the number itself — so subtract 2: 72 − 2 = 70.

Cross-check

Count of proper factors strictly between 1 and the number = total factors − 2 = 72 − 2 = 70. Both 1 and the full product 25 × 33 × 52 are themselves factors, confirming exactly two are removed, leaving 70.