6.45 Let $\varphi(n)$ denote the number of natural numbers less than $n$ that are coprime to $n$.
(1) $p, q$ are two distinct prime numbers. Prove that
$$
\varphi(p q)=(p-1)(q-1) .
$$
(2) Using the result from (1), find the values of $p, q$ that satisfy $\varphi(p q)=3 p+q$. (Anhui Province, Anqing City Junior High School Mathematics Competition, 1991)
[Solution] (1) Since $p, q$ are two distinct prime numbers, the natural numbers less than $p q$ and coprime with $p q$ cannot be divisible by $p$ or $q$.

Among the $p q-1$ natural numbers from 1 to $p q-1$, there are $q-1$ numbers divisible by $p$, and $p-1$ numbers divisible by $q$.
$$
\varphi(p q)=p q-1-(p-1)-(q-1)=(p-1)(q-1) .
$$
(2) From (1), we have
$$
\begin{array}{c}
(p-1)(q-1)=3 p+q \\
p q-4 p-2 q+1=0 \\
(p-2)(q-4)=7 \\
\left\{\left.\begin{array}{l}
p-2=1, \\
q-4=7, \\
p-2=-1, \\
q-4=-7,
\end{array} \right\rvert\,\left\{\begin{array}{l}
p-2=7 \\
q-4=1
\end{array}\right.\right. \\
\left\{\begin{array}{l}
p-2=-7 \\
q-4=-1
\end{array}\right.
\end{array}
$$

Solving, we get $p=3, q=11$.