4-2 Find all positive values of $a$ such that the equation $a^{2} x^{2}+a x+1-7 a^{2}=0$ has two integer roots.
[Solution] If the original equation has two integer roots, then by Vieta's formulas, $-\frac{1}{a}$ is an integer, so $a$ is a number of the form $\frac{1}{n}(n \in \mathbb{N})$. Let $a=\frac{1}{n}$, and transform the original equation into
$$x^{2}+n x+n^{2}-7=0$$

Then $\quad \Delta=n^{2}-4 n^{2}+28=-3 n^{2}+28 \geqslant 0$,
thus $n^{2} \leqslant 9, n \leqslant 3$.
$a=1, \frac{1}{2}$, or $\frac{1}{3}$.
Upon verification, when $a=1, \frac{1}{2}, \frac{1}{3}$, the equation has two integer roots.