$7 \cdot 56$ If there are $n$ integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying the following equation
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=1990 \text {. }
$$

Find the smallest natural number $n(n \geqslant 2)$.
(16th All-Russian Mathematical Olympiad, 1990)