Competition problem — Number theory

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Number theory Difficulty 5.0 AIME, harder Prove it Soviet Union

Problem:

Does there exist a 4-digit integer which cannot be changed into a multiple of 1992 by changing 3 of its digits?

Solution

Solution:

The only 4-digit multiples of $1992$ are: $1992$ , $3984$ , $5976$ , $7968$ , $9960$ . All have first digit odd, second digit $9$ , third digit $>5$ and last digit even, so it is easy to find a number which has all digits different from all of them.

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