【Question 10】
$S(n)$ represents the sum of the digits of the natural number $n$, for example, $S(123)=1+2+3=6$. If two different positive integers $m, n$ satisfy
$$
\left\{\begin{array}{l} 
mn \\
m+S(n)=n+2 S(m)
\end{array} \text {, then we say that } m, n \text { form a pair } \langle m, n>\text { . }\right.
$$

The number of pairs $\langle m, n\rangle$ is $\qquad$ pairs.
【Analysis and Solution】
Number Theory, Congruence; Counting.
We know that a number and the sum of its digits have the same remainder when divided by 9;
That is, $n \equiv S(n)(\bmod 9)$;
Thus, $n-S(n) \equiv 0(\bmod 9)$;
From $m+S(n)=n+2 S(m)$, we get
$$
S(m)=(n-S(n))-(m-S(m)) \equiv 0(\bmod 9) ;
$$

Therefore, $m \equiv 0(\bmod 9)$;
That is, $m$ is a multiple of 9;
Since $m$ has $9+\underbrace{10+10+\cdots+10}_{9 \uparrow 10}=99$ pairs.