Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Prove it United States

Problem:

Determine whether there exists a natural number having exactly $10$ divisors (including itself and $1$ ), each ending in a different digit.

Solution

Solution:

The answer is no.
Suppose that such a number $n$ exists. Since $n$ has a divisor ending in $5$ , $n$ is divisible by $5$ . Then for each divisor $d$ of $n$ not divisible by $5$ , $5d$ also divides $n$ . This contradicts the assumption that $n$ has eight divisors ending in $1,2,3,4,6,7,8$ , or $9$ and only two ending in $5$ or $0$ .

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