### Instruction: What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$?### Response: I notice that the terms in the sum are all of the form $6k + 1$, where $k$ is an integer from $0$ to $43$.
That means the sum is equivalent to $6(0 + 1 + 2 + \cdots + 43) + 44$, where I added $44$ because there are $44$ terms in the sum.
Using the formula for the sum of an arithmetic series, I can simplify this expression to $6 \cdot \frac{43 \cdot 44}{2} + 44$.
Now, to find the remainder when dividing by $6$, I only need to look at the last term, since the first term is a multiple of $6$.
The remainder of dividing $44$ by $6$ is $2$, so that is the answer.