Competition problem — Number theory

Library / /91 of 520

Number theory Difficulty 5.1 AIME, harder Find the answer

11. Find all possible values of $m$ , given that $m$ is a positive integer greater than 2022, and $2022+m$ divides $2022m$ . Find $m$ equals $\qquad$ .

Your answer

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

Solution

【Answer】1011, 2022,
【Analysis】 $\frac{2022 m}{2022+m}$ is an integer, $\frac{2022 m+2022^{2}-2022^{2}}{2022+m}=2022-\frac{2022^{2}}{2022+m}$ is an integer. Therefore, $2022+m$ is a factor of $2022^{2}$ . $2022^{2}=2^{2} \times 3^{2} \times 337^{2}$ , then, $\mathrm{m}$ can be 1011, 2022.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.