2007 AIME I Problems/Problem 1
Contents
[hide]Problem
How many positive perfect squares less than are multiples of ?
Solution
The prime factorization of is . Thus, each square must have at least factors of and factor of and its square root must have factors of and factor of . This means that each square is in the form , where is a positive integer less than . There are solutions.
Solution 2
The perfect squares divisible by are all multiples of : , , , , etc... Since they all have to be less than , or , the closest multiple of to is (), so we know that this is the last term in the sequence. Therefore, we know that there are perfect squares divisible by that are less than .
See also
2007 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
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