1. The number of positive integers $n$ such that $n+1$ divides $n^{2012}+2012$ is $\qquad$ .
From the problem, we know
$$
\begin{array}{l}
n^{2012}+2012 \equiv(-1)^{2012}+2012 \\
=2013 \equiv 0(\bmod n+1) .
\end{array}
$$

Since $2013=3 \times 11 \times 61$, $n$ has $2^{3}-1=7$ solutions.