Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

Find the sum of the reciprocals of all the (positive) divisors of 144.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

As $d$ ranges over the divisors of 144 , so does $144 / d$ , so the sum of $1 / d$ is $1 / 144$ times the sum of the divisors of 144 . Using the formula for the sum of the divisors of a number (or just counting them out by hand), we get that this sum is 403, so the answer is 403/144.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.