### 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, and the rightmost digit is the ones place.
So, $121_{5}$ means $1 \times 5^2 + 2 \times 5^1 + 1 \times 5^0 = 25 + 10 + 1 = 36_{10}$.
Similarly, $11_{5}$ means $1 \times 5^1 + 1 \times 5^0 = 5 + 1 = 6_{10}$.
To find the base five product, I could multiply the decimal values and then convert the result back to base five.
So, $36_{10} \times 6_{10} = 216_{10}$.
To convert $216_{10}$ to base five, I can repeatedly divide by five and write down the remainders.
$216 \div 5 = 43$ with remainder $1$.
$43 \div 5 = 8$ with remainder $3$.
$8 \div 5 = 1$ with remainder $3$.
$1 \div 5 = 0$ with remainder $1$.
So, the base five digits are $1331$, reading from right to left.
Therefore, the base five product is $1331_{5}$.