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# Find the smallest possible integer

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Find the smallest possible integer   such that .

Note: For a real number x,{x} = x - [x] denotes the fractional part of x.

Jun 7, 2021

#1
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Square root is an increasing function, so we only need to find an integer $n\ge100$ such that $\{\sqrt{n-1}\} \le 0.5$ but $\{\sqrt{n}\} >0.5$.  Since $\sqrt{110} = 10.488+$ and $\sqrt{111} = 10.535+$ so the smallest such $n$ is 111.

Jun 8, 2021
#2
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Thank you!

Guest Jun 10, 2021