Problem:
A positive integer is nice if there is a positive integer with exactly four positive divisors (including and ) such that the sum of the four divisors is equal to . How many numbers in the set are nice?
Answer Choices:
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Solution:
Let denote a nice number from the given set. An integer has exactly four divisors if and only if or , where and (with ) are prime numbers. In the former case, the sum of the four divisor is equal to . Note that and . Therefore we must have and . Because is odd, must be an even number. If , then must be divisible by . In the given set only and satisfy these requirements. However neither nor are prime. If is odd, then must be divisible by . In the given set, only and are divisible by . None of the pairs of factors of , namely , gives rise to primes and . This leaves , which is the only nice number in the given set.
Remark: Note that is nice in five ways. The other four ways are , and .
The problems on this page are the property of the MAA's American Mathematics Competitions