11. (1980 Leningrad Mathematical Olympiad) Given the equation $x^{4}-p x^{3}+q=0$ has an integer root, find the prime $p$ and $q$.

 untranslated text remains the same as requested.
11. Let the integer $x$ satisfy
$$
q=x^{3}(p-x) \text {, }
$$

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

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
$$
p=q+1 \text {. }
$$

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