Problem 890

Three, Solution: If $2 \mid p q$, without loss of generality, let $p=2$, then $2 q \mid 5^{2}+5^{q}$, so $q \mid 5^{q}+25$.

By Fermat's Little Theorem, $q \mid 5^{q}-5$, thus $q \mid 30$, i.e., $q=2$, 3, 5. It is easy to verify that the prime pair $(2,2)$ does not meet the requirements, while $(2,3)$ and $(2,5)$ do meet the requirements.

If $p q$ is odd and $5 \mid p q$, without loss of generality, let $p=5$, then $5 q \mid 5^{5}+5^{q}$, so $q \mid 5^{q-1}+625$.

When $q=5$, the prime pair $(5,5)$ meets the requirements. When $q \neq 5$, by Fermat's Little Theorem, we have $q \mid 5^{q-1}-1$, hence $q \mid 626$. Since $q$ is an odd prime, and the only odd prime factor of 626 is 313, so $q=313$. After verification, the prime pair $(5,313)$ meets the requirements.

If $p, q$ are neither 2 nor 5, then $p q \mid 5^{p-1}+5^{q-1}$, so
$$5^{p-1}+5^{q-1} \equiv 0(\bmod p).$$

By Fermat's Little Theorem, we get $5^{p-1} \equiv 1(\bmod p)$,
so from (1) and (2) we have
$$5^{q-1} \equiv-1(\bmod p).$$

Let $p-1=2^{k}(2 r-1), q-1=2^{l}(2 s-1)$, where $k$, $l, r, s$ are positive integers.
If $k \leqslant l$, then from (2), (3) it is easy to see
$$1=1^{2^{l-k}(2 s-1)} \equiv\left(5^{p-1}\right)^{2^{l-k}(2 s-1)}=5_{l}^{2(2 \Gamma-1)(2 s-1)}=$$
$\left(5^{q-1}\right)^{2 r-1} \equiv(-1)^{2 r-1} \equiv-1(\bmod p)$,
which contradicts $p \neq 2$! So $k>l$.
Similarly, we have $k<l$, which is a contradiction! Therefore, there are no $(p, q)$ that meet the requirements in this case.

In summary, all prime pairs $(p, q)$ that meet the requirements are
$$(2,3),(3,2),(2,5),(5,2),(5,5),(5,313) \text { and }$$