Find the smallest positive integer $n$ that satisfies $$\lfloor \sqrt{n} \rfloor = \lfloor \sqrt{n + 34} \rfloor$$
2026-03-27 15:58:59.1774627139
Floor equation with square roots
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The trick here is to see when the difference between two squares is equal to 34. $17^2-16^2=289-256=33$ and $18^2-17^2=324-289=35$. Therefore n should be around $289$, and check it by plugging it in: $\lfloor{289}\rfloor=\lfloor{323}\rfloor$ and $324=18^2$ so $\boxed{289}$ is valid.