$$\begin{aligned} & m=6 a, n=6 b \\ & \text { So } \operatorname{gcd}(m, n)=6 \Rightarrow \operatorname{gcd}(a, b)=1 \\ & m=6 a \geq 10 \Rightarrow a \geq\left[\frac{10}{6}\right]=2 \\ & m=6 a \leq 99 \Rightarrow a \leq\left[\frac{99}{6}\right]=16 \end{aligned}$$

So $a, b \in\{2,3, \ldots, 16\}$, and we count how many coprime pairs $(a, b)$ with $a< b, \operatorname{gcd}(a, b)=1$

$$\begin{aligned} & a=2 \Rightarrow b=3,5,7,9,11,13,15 \Rightarrow 7 \\ & a=3 \Rightarrow \mathrm{~b}=4,5,7,8,10,11,13,14,16 \Rightarrow 9 \\ & a=4 \Rightarrow b=5,7,9,11,13,15 \Rightarrow 6 \\ & a=5 \Rightarrow \mathrm{~b}=6,7,8,9,11,12,13,14,16 \Rightarrow 9 \\ & a=6 \Rightarrow \mathrm{~b}=7,11,13 \Rightarrow 3 \\ & a=7 \Rightarrow b=8,9,10,11,12,13,15,16 \Rightarrow 8 \\ & a=8 \Rightarrow \mathrm{~b}=9,11,13,15 \Rightarrow 4 \\ & a=9 \Rightarrow \mathrm{~b}=10,11,13,14,16 \Rightarrow 5 \\ & a=10 \Rightarrow b=11,13 \Rightarrow 2 \\ & a=11 \Rightarrow b=12,13,14,15,16 \Rightarrow 5 \\ & a=12 \Rightarrow b=13,17 \times \rightarrow \text { only } 13 \text { is valid } \Rightarrow 1 \\ & a=13 \Rightarrow b=14,15,16 \Rightarrow 3 \\ & a=14 \Rightarrow b=15, \Rightarrow 1 \\ & a=15 \Rightarrow b=16 \Rightarrow 1 \end{aligned}$$

$\begin{aligned} & \text { Total }=7+9+6+9+3+8+4+5+2+5+1 \\ & +3+1+1=64\end{aligned}$