AIME 2017 I · 第 3 题

AIME 2017 I — Problem 3

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem 3

For a positive integer $n$ , let $d_n$ be the units digit of $1 + 2 + \dots + n$ . Find the remainder when

$\sum_{n=1}^{2017} d_n$ is divided by $1000$ .

解析

Solution

We see that $d_n$ appears in cycles of $20$ and the cycles are

$1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0,$ adding a total of $70$ each cycle. Since $\left\lfloor\frac{2017}{20}\right\rfloor=100$ , we know that by $2017$ , there have been $100$ cycles and $7000$ has been added. This can be discarded as we're just looking for the last three digits. Adding up the first $17$ of the cycle of $20$ , we can see that the answer is $\boxed{069}$ .

~ Maths_Is_Hard

Video Solution

https://youtu.be/BiiKzctXDJg ~ Shrea S