### Instruction: What is the base five product of the numbers $121_{5}$ and $11_{5}$?### Response: I know that base five means that each digit represents a power of five, starting from the rightmost one as the zeroth power.
So, to convert a base five number to base ten, I can multiply each digit by its corresponding power of five and add them up.
For example, $121_{5} = 1 \cdot 5^{2} + 2 \cdot 5^{1} + 1 \cdot 5^{0} = 25 + 10 + 1 = 36_{10}$.
Similarly, $11_{5} = 1 \cdot 5^{1} + 1 \cdot 5^{0} = 5 + 1 = 6_{10}$.
So, in base ten, the product of these two numbers is $36_{10} \times 6_{10} = 216_{10}$.
But I need to find the base five product, not the base ten product.
So, I need to convert $216_{10}$ back to base five.
To do that, I can repeatedly divide by five and write down the remainders.
So, $216 \div 5 = 43$ with a remainder of $1$.
Then, $43 \div 5 = 8$ with a remainder of $3$.
Then, $8 \div 5 = 1$ with a remainder of $3$.
And finally, $1 \div 5 = 0$ with a remainder of $1$.
So, the base five representation of $216$ is the remainders read from bottom to top: $1331_{5}$.
Therefore, the base five product of $121_{5}$ and $11_{5}$ is $1331_{5}$.