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Harvard-MIT November Tournament

Number theory Difficulty 4.2 AIME Find the answer United States

Problem:
Determine the remainder when $1+2+\cdots+2014$ is divided by $2012$ .

Your answer

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

Solution

Solution:
We wish to find the value of $1+2+\cdots+2014$ modulo $2012$ . We have
$1+2+\cdots+2014=\frac{1}{2}(2014)(2015)=1007 \cdot 2015 \equiv 1007 \cdot 3=3021 \equiv 1009 \pmod{2012}$

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