What is the unit digit of the number $3^{1}+3^{2}+3^{3}+3^{4}+\cdots 3^{2013}$?

Remember, for example, that $3^{2}=3 \times 3=9$. And $3^{3}=3 \times 3 \times 3=27$. Also remember that the unit digit of a number is the last one on the right. For example, the unit digit of 376564534539 is 9.
Solution

## Note that

$3^{1}=3$, whose unit digit is 3.

$3^{2}=9$, whose unit digit is 9.

$3^{3}=27$, whose unit digit is 7.

$3^{4}=81$, whose unit digit is 1.

$3^{5}=243$, whose unit digit is 3.

$3^{6}=729$, whose unit digit is 9.

$3^{7}=2187$, whose unit digit is 7.

$3^{8}=6561$, whose unit digit is 1.

We can already notice that every four powers, the unit digits repeat (always 3, 9, 7, and 1, in that order). And also that, since $3+7=10$ and $1+9=10$, these groups of four numbers do not affect the unit digit, because adding a number that ends in zero does not change the unit digit.

Dividing 2013 by 4, we notice that the remainder is 1. Therefore, the unit digit of $3^{1}+3^{2}+3^{3}+3^{4}+\cdots+3^{2013}$ is 3.