2020 AIME II Problems/Problem 1

Revision as of 20:38, 7 June 2020 by CoolCarsOnTheRun (talk | contribs) (Solution)

Problem

Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.

Solution

First, we find the prime factorization of $20^{20}$, which is $2^{40}\times5^{20}$. The equation ${m^2n = 20 ^{20}}$ tells us that we want to select a perfect square factor of $20^{20}$, $m^2$. $n$ will be assigned by default. There are $21\times11=231$ ways to select a perfect square factor of $20^{20}$, thus our answer is $\mbox{231}$.

~superagh

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