Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Prove it Philippines

Problem:
Find all values of integers $x$ and $y$ satisfying $2^{3x} + 5^{3y} = 189$ .

Solution

Solution:
We have
$\begin{aligned} 2^{3x} + 5^{3y} &= \left(2^x + 5^y\right)\left(2^{2x} - 2^x 5^y + 5^{2y}\right) \\ &= (9)(21) \\ &= 189 \end{aligned}$
Clearly, $(x, y) = (2, 1)$ is the only solution pair to the equation.

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