1. Find all positive integer triples $(I, M, C)$ (where $I$, $M$, and $C$ are prime numbers and $I \leqslant M \leqslant C$) such that
$$
I \cdot M \cdot C=I+M+C+1007 .
$$
1. $(2,2,337)$.

If $I$, $M$, and $C$ are all odd, then $I \cdot M \cdot C$ is odd while $I+M+C+1007$ is even, which is a contradiction.
Therefore, $I$, $M$, and $C$ must include an even number, and since they are all prime numbers,
according to the size relationship, we get $I=2$.
Substituting into equation (1) yields
$$
\begin{array}{l}
2 M \cdot C - M - C = 1009 \\
\Rightarrow \quad (2 M - 1)(2 C - 1) = 2019 \\
\quad = 1 \times 2019 = 3 \times 673
\end{array}
$$
$\Rightarrow$ The prime solution is $(M, C) = (2, 337)$.
Thus, $(2,2,337)$ is the required prime solution.