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**Unbeatable0** Therefore if $\displaystyle (x,y,z)$ is a solution then so is $\displaystyle \left(\frac{x}{7},\frac{y}{7},\frac{z}{7}\right)$, and applying again and again:

$\displaystyle \left(\frac{x}{7^n},\frac{y}{7^n},\frac{z}{7^n}\ri ght)$

is a soloution for all $\displaystyle n\in\mathbb{N}$, and in particular made of integers.

But it's impossible that these numbers are integers for all $\displaystyle n$ if one of $\displaystyle x,y,z$ is nonzero, so we have the only solution: $\displaystyle (0,0,0)$