Example 2 Let $72 \mid \overline{a 679 b}$. Find the values of $a$ and $b$. $(1980$, Canadian Mathematics Competition)
Note that, $72=8 \times 9, (8,9)=1$.
Therefore, we only need to discuss the values of $a$ and $b$ when both 8 and 9 divide $\bar{a} \overline{7} 9 b$.
By $8|\overline{a 679 b} \Rightarrow 8| \overline{79 b} \Rightarrow b=2$;
By $9|\overline{a 679 b} \Rightarrow 9|(a+6+7+9+b)$
$$
\Rightarrow 9 \text { | }(a+6+7+9+2) \Rightarrow a=3 \text {. }
$$