2005 AIME I Problems/Problem 3
Problem
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?
Solution (Basic Casework and Combinations)
Suppose is such an integer. Because has proper divisors, it must have divisors,, so must be in the form or for distinct prime numbers and .
In the first case, the three proper divisors of are , and . Thus, we need to pick two prime numbers less than . There are fifteen of these ( and ) so there are ways to choose a pair of primes from the list and thus numbers of the first type.
In the second case, the three proper divisors of are 1, and . Thus we need to pick a prime number whose square is less than . There are four of these ( and ) and so four numbers of the second type.
Thus there are integers that meet the given conditions.
~lpieleanu (Minor editing)
~ rollover2020 (extremely minor editing)
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 4 | |
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