Find the total number of factors (divisors) of 2500 × 775.

Concept

Any whole number can be written as a product of prime powers n = pa qb rc … The total number of positive divisors of n is the product of one-more-than each exponent: (a + 1)(b + 1)(c + 1) … Every distinct prime contributes its own factor, so each prime base must be separated and each exponent must be increased by 1 before multiplying.

Application

1. Factorise each part into primes: 2500 = 22 × 54 and 775 = (7 × 11)5 = 75 × 115 (note 77 is not prime — it splits into 7 and 11).

2. Combine: 2500 × 775 = 22 × 54 × 75 × 115 — four distinct primes with exponents 2, 4, 5 and 5.

3. Add 1 to each exponent and multiply: (2 + 1)(4 + 1)(5 + 1)(5 + 1) = 3 × 5 × 6 × 6 = 540.

Cross-check

Treating the two equal exponents together: 3 × 5 × 62 = 15 × 36 = 540, the same value — and all four primes 2, 5, 7, 11 have been counted exactly once. The number of factors is 540.