Singapur, 2015

Number theory Difficulty 4.9 AIME Prove it Singapore

Let $n = \overline{30x070y03}$ be a 9-digit integer. Find all possible values of the pair $(x, y)$ , so that $n$ is a multiple of 37.

Solution

We have
$n = 300070003 + 10^6x + 10^2y = 37(8110000 + 27027x + 3y) + (3 + x - 11y).$
Since $0 \le x, y \le 9$ , we have $-96 \le 3 + x - 11y \le 12$ . Also $37 \mid 3 + x - 11y$ . Thus $3 + x - 11y = 0, -37$ or $-74$ and we get $(x, y) = (8, 1), (4, 4), (0, 7)$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.