Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Prove it Saudi Arabia

Prove that there are no positive integers $x$ , $y$ , $z$ such that
$(3x + 4y)(4x + 5y) = 7^{z}$

Solution

From the condition, we can see that both numbers $3x + 4y$ and $4x + 5y$ are powers of $7$ , so such must be also their division. However
$1 < \frac{4x + 5y}{3x + 4y} < 2$
and can't be power of $7$ .

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