How many numbers from 1 to 400 have exactly three factors (divisors)?

Concept

For a positive integer, the divisor-counting rule says to add 1 to every exponent in its prime factorization and multiply those results. A number has exactly three positive divisors only when it has one prime factor with exponent 2.

Thus such a number has the form n = p2, where p is prime; its divisors are 1, p, and p2.

Application

1. Require p2 ≤ 400 so that the prime square lies in the stated range.

2. Taking nonnegative square roots gives p ≤ 20.

3. The primes at most 20 are 2, 3, 5, 7, 11, 13, 17, and 19; this list contains eight primes.

Cross-check

The boundary is exact: 192 = 361 ≤ 400, whereas the next prime gives 232 = 529 > 400.

Therefore, 8 numbers from 1 to 400 have exactly three divisors.