### Instruction: What is the base five product of the numbers $121_{5}$ and $11_{5}$?### Response: To multiply two numbers in base five, I can use the same algorithm as in base ten, but with different digits and carry-overs.
I'll start by multiplying $121_{5}$ by the ones digit of $11_{5}$, which is one, so the product is just $121_{5}$.
Then I'll multiply $121_{5}$ by the fives digit of $11_{5}$, which is one, and shift the product one place to the left, so it becomes $1210_{5}$.
To add these two products, I'll align them and add the digits in each column, starting from the right.
In the ones column, I have $1 + 0 = 1$, so I write one below the line.
In the fives column, I have $2 + 1 = 3$, so I write three below the line.
In the twenty-fives column, I have $1 + 2 = 3$, so I write three below the line.
In the one hundred twenty-fives column, I have $0 + 1 = 1$, so I write one below the line.
There are no more digits to add, so I'm done.
The base five product of $121_{5}$ and $11_{5}$ is $1331_{5}$.