2020 AIME II Problems/Problem 1
Find the number of ordered pairs of positive integers such that .
First, we find the prime factorization of , which is . The equation tells us that we want to select a perfect square factor of , . The might throw you off here, but it's actually kind of irrelevant because once is selected, the remaining factor will already be assigned as . There are ways to select a perfect square factor of , thus our answer is .
Solution 2 (Official MAA)
Because , if , there must be nonnegative integers , , , and such that and . Then and The first equation has solutions corresponding to , and the second equation has solutions corresponding to . Therefore there are a total of ordered pairs such that .
Video Solution 2
Purple Comet Math Meet April 2020
Notice, that this was the exact same problem (with different wording of course) as Purple Comet HS problem 3 and remembering the answer, put .
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