1. [5] Determine the remainder when $1+2+\cdots+2014$ is divided by 2012 .
Answer: 1009 We wish to find the value of $1+2+\cdots+2014$ modulo 2012 . We have
$$
1+2+\cdots+2014=\frac{1}{2}(2014)(2015)=1007 \cdot 2015 \equiv 1007 \cdot 3=3021 \equiv 1009 \quad(\bmod 2012)
$$