2005 AIME I Problems/Problem 3

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Problem

How many positive integers have exactly three proper divisors, each of which is less than 50?

Solution

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The following solution contains errors; please correct it: Having three proper divisors means that there are 4 regular divisors. So the number can be written as $\displaystyle p_{1}p_{2}$ where $\displaystyle p_{1}$ and $\displaystyle p_{2}$ are primes. The primes under fifty are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. There are 15 of them. So there are ${15 \choose 2} =105$ such numbers.

See also

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