### Instruction: Convert $222_{10}$ to base $13$.  Use digits A, B, and C if necessary.### Response: To convert $222_{10}$ to base $13$, I need to find the largest power of $13$ that is less than or equal to $222_{10}$.
I know that $13^2 = 169$, which is less than $222_{10}$, but $13^3 = 2197$, which is too big.
So the largest power of $13$ that I need is $13^2$.
I can divide $222_{10}$ by $13^2$ to get the quotient and the remainder.
Using long division, I get $222_{10} \div 13^2 = 1 R 53$.
This means that the coefficient of $13^2$ in the base $13$ expansion of $222_{10}$ is $1$.
I can write $222_{10} = 1 \cdot 13^2 + 53$.
Now I need to repeat the same process for the remainder $53$.
I need to find the largest power of $13$ that is less than or equal to $53$.
I know that $13^1 = 13$, which is less than $53$, but $13^2 = 169$, which is too big.
So the largest power of $13$ that I need is $13^1$.
I can divide $53$ by $13^1$ to get the quotient and the remainder.
Using long division, I get $53 \div 13 = 4 R 1$.
This means that the coefficient of $13^1$ in the base $13$ expansion of $53$ is $4$.
I can write $53 = 4 \cdot 13^1 + 1$.
Now I need to repeat the same process for the remainder $1$.
I need to find the largest power of $13$ that is less than or equal to $1$.
I know that $13^0 = 1$, which is equal to $1$.
So the largest power of $13$ that I need is $13^0$.
I can divide $1$ by $13^0$ to get the quotient and the remainder.
Using long division, I get $1 \div 13^0 = 1 R 0$.
This means that the coefficient of $13^0$ in the base $13$ expansion of $1$ is $1$.
I can write $1 = 1 \cdot 13^0 + 0$.
Now I have all the coefficients of the powers of $13$ that make up $222_{10}$.
I can write them in descending order of the exponents, with a subscript of $13$ to indicate the base.
I get $222_{10} = 1 \cdot 13^2 + 4 \cdot 13^1 + 1 \cdot 13^0$.
This is equivalent to $222_{10} = (141)_{13}$.
I don't need to use any digits A, B, or C, because all the coefficients are less than $10$.