jmc

To find the base $5$ representation of $123_{10}$, we first write $123$ as the sum of powers of $5$. The largest power of $5$ that is less than $123$ is $5^2=25$ and the largest multiple of $25$ that is less than $123$ is $4\cdot 25=100$. Thus, we have $123=4\cdot 25+23$. We then consider the largest power of $5$ that is less than $23$ which is $5^1=5$. The largest multiple of $5$ that is less than $23$ is $4\cdot 5=20$ and we have $23-20=3$, which can be written as $3\cdot 5^0$. Therefore, we can write $123$ as $$123=4\cdot 5^2+4\cdot 5^1+3\cdot 5^0.$$Thus, $123_{10}$ in base $5$ is $\boxed{443_5}$.