, 2019

Number theory Difficulty 5.2 AIME, harder Prove it Netherlands

If you try to divide the number $19$ by $5$ , you will get a remainder. The number $5$ fits $3$ times in $19$ and you will be left with $4$ as remainder. There are two positive integers $n$ having the following property: if you divide $n^2$ by $2n + 1$ , you will get a remainder of $1000$ . What are these two integers?

Solution

$666$ and $1999$

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