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max product a x b , max no of common divisors

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E10

December 12, 2022
number theorydivides

Problem Statement

The pair of positive integers (a,b)(a, b) is such that a does not divide bb, bb does not divide a, both numbers are at most 100100, and they have the maximal possible number of common divisors. What is the largest possible value of a⋅⋅ba \cdot· b?
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