Addition in base seven

Statement

Compute $(345)_7 + (256)_7$ working directly in base seven.

Solution

Add the digits column by column, carrying whenever the total reaches $7$: $\begin{aligned} 5+6 &= 11 = 1\cdot 7 + 4 &&\Rightarrow \text{digit } 4,\ \text{carry } 1, \ 4+5+1 &= 10 = 1\cdot 7 + 3 &&\Rightarrow \text{digit } 3,\ \text{carry } 1, \ 3+2+1 &= 6 &&\Rightarrow \text{digit } 6. \end{aligned}$ So $(345)_7 + (256)_7 = \boxed{(634)_7}$. Check in base ten: $180 + 139 = 319 = (634)_7$.