## Subject 3

a) $x+(x+1)+(x+2)+\ldots+(x+6)=\overline{1 a 091}$ $1 p$

$$
7 x+21=\overline{1 a 091}
$$

$1 p$
$7 x=\overline{1 a 070} \Rightarrow x: 10$ ..... $.1 p$
$\overline{1 a 070} \vdots 7 \Rightarrow a=4, x=2010 ; \ldots ; x+6=2016$ ..... $1 p$
b) $n=2^{2014}-1$ ..... $1 p$
$u\left(2^{2014}\right)=4$ ..... $1 p$
$u(n)=4-1=3$ ..... $1 p$
## Subject 3

a) The sum of seven consecutive natural numbers is $\overline{1 a 091}$. Find the seven numbers.

Prof. Daniela Bădulescu

b) Calculate the last digit of the natural number $n=1+2+2^{2}+2^{3}+\ldots .+2^{2013}$.

Prof. Amelia Duca