Competition problem — Number theory

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Number theory Difficulty 4.4 AIME Find the answer United States

Problem:

Albert writes down all of the multiples of $9$ between $9$ and $999$ , inclusive. Compute the sum of the digits he wrote.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

If $x$ is a multiple of $9$ , so is $999 - x$ , and their digits always sum to $27$ . The $112$ multiples from $0$ to $999$ form $56$ such pairs, so the sum of their digits is $56 \cdot 27 = 1512$ .

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