### Instruction: What is $123_{10}$ when expressed in base $5$?### Response: To convert a number from base $10$ to base $5$, I need to divide it repeatedly by $5$ and keep track of the remainders.
The remainders will form the digits of the base $5$ number, starting from the rightmost one.
So, I start by dividing $123$ by $5$.
I get $24$ with a remainder of $3$.
This means that the rightmost digit of the base $5$ number is $3$.
Next, I divide $24$ by $5$.
I get $4$ with a remainder of $4$.
This means that the second digit from the right of the base $5$ number is $4$.
Then, I divide $4$ by $5$.
I get $0$ with a remainder of $4$.
This means that the third digit from the right of the base $5$ number is also $4$.
Since the quotient is $0$, I stop the division process.
I have found all the digits of the base $5$ number, and they are $443$.