23. $[\mathbf{1 2}]$ Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9 , inclusive, and $N$ is a 7 -digit positive integer. If $N$ is divisible by 792 , determine all possible ordered triples $(A, B, C)$.
Answer: $(0,5,5),(4,5,1),(6,4,9)$ First, note that $792=2^{3} \times 3^{2} \times 11$. So we get that
$$
\begin{array}{c}
8|N \Rightarrow 8| \overline{7 C 2} \Rightarrow 8 \mid 10 C+6 \Rightarrow C=1,5,9 \\
9|N \Rightarrow 9| 5+A+B+3+7+C+2 \Rightarrow A+B+C=1,10,19 \\
11|N \Rightarrow 11| 5-A+B-3+7-C+2 \Rightarrow-A+B-C=-11,0
\end{array}
$$

Adding the last two equations, and noting that they sum to $2 B$, which must be even, we get that $B=4,5$.
Checking values of $C$ we get possible triplets of $(0,5,5),(4,5,1)$, and $(6,4,9)$.