What is the largest value of $n$ for which $2n + 1$ is a factor of $122 + n^{2}$? Given that $n$ is a natural number, what is its largest value such that  $2n + 1$ is a factor of  $122 + n^{2}$?

Since $2n+1$ is odd, it divides $n^2+122$ iff it divides $4n^2+488$ and
$$
4n^2+488=(2n+1)(2n-1)+489
$$
Thus, $2n+1\mid489$. The largest such $n$ would be $244$ and if we check
$$
2\cdot244+1\mid244^2+122
$$
Thus, $n=244$ is the largest $n$ so that $2n+1\mid ^2+122$.
