Competition problem — Number theory

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Number theory Difficulty 4.8 AIME Find the answer

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 $(A, B, C)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

First, note that $792=2^{3} \times 3^{2} \times 11$ . So we get that $\begin{gathered} 8|N \Rightarrow 8| \overline{7 C 2} \Rightarrow 8 \mid 10 C+6 \Rightarrow C=1,5,9 \\ 9|N \Rightarrow 9| 5+A+B+3+7+C+2 \Rightarrow A+B+C=1,10,19 \\ 11|N \Rightarrow 11| 5-A+B-3+7-C+2 \Rightarrow-A+B-C=-11,0 \end{gathered}$ 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 $(6,4,9)$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.