Analyzing $biggllfloor{frac{x}{5}}biggrfloor=bigglfloor{frac{x}{7}}biggrfloor$

Rewrite the given condition as $$5nle x<5n+5,quad 7nle x<7n+7$$ where $n$ is an integer. Since $xge0$, $nge0$ as well, so the above is equivalent to $7nle x<5n+5$. The only nonnegative integers $n$ satisfying $7n<5n+5$ are $0$, $1$, and $2$. And so just a little cleanup is needed.

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