A number n selected from 1,,1000.

Question

A number $n$ selected from $\{1,\dots,1000\}$. Find the probability that $\dfrac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer.

Accepted Answer

$$\frac{\sum i^2}{\sum i} = \frac{2n+1}{3}$$

This is an integer when $2n+1\equiv0\pmod3$, i.e. $n\equiv1\pmod3$. Values of $n$ from 1 to 1000: $n=1,4,7,\dots,1000$ --- 334 values.

$$P = \frac{334}{1000}=0.334$$

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