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Girls in Math at Yale 2022 Problem 10: Dividing 15^6

Source:

February 27, 2022
Yalecollege

Problem Statement

How many ways are there to choose distinct positive integers a,b,c,da, b, c, d dividing 15615^6 such that none of a,b,c,a, b, c, or dd divide each other? (Order does not matter.)
Proposed by Miles Yamner and Andrew Wu
(Note: wording changed from original to clarify)
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