Problem 1. a) Find the last digit of the number $x=2^{0}+2^{1}+2^{2}+\ldots+2^{2016}$ b) Let $n=1+3+5+\cdots+2015$. Show that $n$ is a perfect square.
## Solution:

a) $\operatorname{uc}\left(2^{n}\right) \in\{2 ; 4 ; 6 ; 8\}, n \in \mathbb{N}^{*}$

$1 p$

![](https://cdn.mathpix.com/cropped/2024_06_07_46238a177f184c489daag-1.jpg?height=68&width=1619&top_left_y=791&top_left_x=272)

$u c\left(2^{1}+2^{2}+2^{3}+2^{4}\right)=\operatorname{uc}\left(2^{5}+2^{6}+2^{7}+2^{8}\right)=\ldots=u c\left(2^{2013}+2^{2014}+2^{2015}+2^{2016}\right)=0 \ldots \ldots 1 p$

$u c\left(2^{0}\right)=1 \Rightarrow \operatorname{uc}(x)=1$

$1 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_46238a177f184c489daag-1.jpg?height=66&width=1684&top_left_y=943&top_left_x=208)

Completion, $\mathrm{n}=1008^{2}$