Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it Belarus

Find all possible values of digits $a$ and $b$ such that
$(ab)^3 = \overline{(a-3)(b-3)(b+2)(a+2)ab}.$
(As usual, by $\overline{xyz}$ we denote an integer number, which decimal representation consists of digits $x, y, \dots, z$ in that order.)

Solution

$a = 7, b = 6$ .

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