When the base-16 integer $729_{16}$ is divided by the base 10 number $15$, what is the remainder? Express your answer in base 10.
To find the remainder, I need to convert the base-16 integer to base 10 first. I can do this by using the place value of each digit and multiplying it by the corresponding power of 16. For example, the rightmost digit is 9, and it has a place value of $16^0 = 1$, so it contributes $9 \times 1 = 9$ to the base 10 value. The next digit is 2, and it has a place value of $16^1 = 16$, so it contributes $2 \times 16 = 32$ to the base 10 value. The leftmost digit is 7, and it has a place value of $16^2 = 256$, so it contributes $7 \times 256 = 1792$ to the base 10 value. Adding these contributions together, I get $9 + 32 + 1792 = 1833$ as the base 10 equivalent of $729_{16}$. Now I can divide 1833 by 15 and find the remainder using long division or a calculator. I get 1833 divided by 15 equals 122 with a remainder of 3. This means that the base 10 remainder is 3. 