### Instruction: What is the last digit of the base $6$ representation of the base ten integer $355$?### Response: To convert a number from base ten to base six, I need to repeatedly divide by six and write down the remainders until I reach zero.
The last digit of the base six representation will be the first remainder I get.
So I start by dividing $355$ by $6$.
I can use long division or a calculator to find that $355 \div 6 = 59$ with a remainder of $1$.
That means the last digit of the base six representation is $1$.
I can check this by multiplying $1$ by $6^0$, which is $1$, and seeing that it matches the remainder.