3.5 Given the equation $x^{4}-p x^{3}+q=0$ has an integer root, find the prime numbers $p$ and $q$.
(Leningrad Mathematical Olympiad, 1980)
［Solution］Let the integer $x$ satisfy
$$
q=x^{3}(p-x),
$$

then $\quad x \mid q$.
Since $q$ is a prime, it can only be that
$$
x= \pm 1, \quad x= \pm q .
$$

When $x= \pm q$, from (1) we have
$$
q^{2} \mid 1,
$$

which is impossible (since $q$ is a prime).
When $x=-1$, we have
$$
p+q+1=0,
$$

which is also impossible.
When $x=1$, from (1) we get
$$
p=q+1 .
$$

Since $p$ and $q$ are both primes, then $p=3, q=2$.