9.17 Let $\overline{a 679 b}$ be a five-digit number in decimal notation, divisible by 72. Determine the values of $a$ and $b$.
(12th Canadian Mathematics Competition, 1980)
[Solution] Since $72=8 \cdot 9, 8$ and 9 are coprime.
Therefore, if $\overline{a 679 b}$ is divisible by 72, it must be divisible by both 8 and 9.
If $\overline{a 679 b}$ is divisible by 8, the necessary and sufficient condition is that the last three digits $79 b$ are divisible by 8.
This yields the unique $b=2$.
If $\overline{a 6792}$ is divisible by 9, the necessary and sufficient condition is that the sum of its digits is divisible by 9. That is,
$$
a+6+7+9+2=24+a
$$

is divisible by 9, yielding the unique $a=3$.
Thus,
$$
a=3, \quad b=2,
$$

and at this point
$$
36792=72 \cdot 511 \text {. }
$$