2010 AIME I Problems/Problem 1
Contents
Problem
Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let be the probability that exactly one of the selected divisors is a perfect square. The probability can be expressed in the form , where and are relatively prime positive integers. Find .
Solution 1
. Thus there are divisors, of which are squares (the exponent of each prime factor must either be or ). Therefore the probability is
Solution 2 (Using a Bit More Counting)
The prime factorization of is . Therefore, the number of divisors of is or , of which are perfect squares. The number of ways we can choose perfect square from the two distinct divisors is . The total number of ways to pick two divisors is
Thus, the probability is
Video Solution
https://www.youtube.com/watch?v=YJeF9dLJZuw (Osman Nal)
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
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