Competition problem — Number theory

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Number theory Difficulty 5.0 AIME, harder Prove it Estonia

Does there exist a positive integer $n$ such that across all representations $n = ab$ , the digits $0, 1, \dots, 9$ are all present as the final digit of $a^b$ at least once?

Solution

Answer: No.

If $n$ is odd, both $a$ and $b$ will always be odd, but then $a^b$ will never end with a $0$ . If $n$ is even, then $a$ or $b$ will be even, so $a^b$ will be even or a perfect square. In either case, it will never end in a $7$ .

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.