Track / Stage 5 / 307 of 400 #907 of 2000

Problem 907

AIME late

Number theory Difficulty 5.8 Prove it

In the decimal representation of a certain number, the digits are arranged from left to right in descending order. Can this number be a multiple of 111?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Suppose that the numbers indicated in the condition and divisible by 111 exist. Let $A$ be the smallest among them. There are two cases.

1) The number $A$ ends in zero. By erasing it, we obtain a smaller number with digits in descending order, also divisible by 111. Contradiction.
2) $A$ ends in a non-zero digit. Then the number $A-111$ is also divisible by 111 and the digits in its representation are in descending order. Again, a contradiction.

## Answer

It cannot.

## Problem

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