Competition problem — Number theory

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Number theory Difficulty 5.1 AIME, harder Prove it Belarus

Let a six-digit number $A$ be divisible by $101$ .
Prove that one can transpose two digits of $A$ so that the obtained number is also divisible by $101$ .

Solution

It is possible to transpose either $a_5$ and $a_1$ or $a_4$ and $a_0$ to obtain the number which is divisible by $101$ .

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