Problem:
How many positive integers have exactly three proper divisors, each of which is less than ? (A proper divisor of a positive integer is a positive integer divisor of other than itself.)
Solution:
There are two types of integers that have three proper divisors. If , where and are distinct primes, then the three proper divisors of are , and ; and if , where is a prime, then the three proper divisors of are , and . Because there are prime numbers less than , there are integers of the first type. There are integers of the second type because , and are the only primes with squares less than . Thus there are integers that meet the given conditions.
The problems on this page are the property of the MAA's American Mathematics Competitions