2005 AIME I Problems/Problem 3
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Problem
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?
Solution
Suppose 
is such an integer. Then one of its factors is 
, 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 
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.
See also
| 2005 AIME I (Problems • Resources) | ||
| Preceded by Problem 2 | Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
