Competition problem — Number theory

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Number theory Difficulty 3.9 AMC 10/12 Find the answer United States

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.