Competition problem — Number theory

Library / /597 of 1394

Number theory Difficulty 5.2 AIME, harder Prove it United States

Problem:

For any positive integer $a$ , let $\tau(a)$ be the number of positive divisors of $a$ . Find, with proof, the largest possible value of $4 \tau(n)-n$ over all positive integers $n$ .

Solution

Solution:

Let $d$ be the number of divisors of $n$ less than or equal to $\frac{n}{4}$ . Then, $\tau(n)-3 \leq d \leq \frac{n}{4} \Longrightarrow$ $4 \tau(n)-n \leq 12$ . We claim the answer is $12$ . This is achieved by $n=12$ .

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