2020 AIME II Problems/Problem 1
Problem
Find the number of ordered pairs of positive integers such that .
Solution
First, we find the prime factorization of , which is . The equation tells us that we want to select a perfect square factor of , . will be assigned by default. There are ways to select a perfect square factor of , thus our answer is .
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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
https://www.youtube.com/watch?v=x0QznvXcwHY
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See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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