### Instruction: In the base-three number system, $a = 2012_3$ and $b = 201_3.$ What is the product $ab$ expressed in the base-three number system?### Response: I know that in the base-three number system, each digit represents a power of three, so I can write $a$ and $b$ as sums of powers of three.
For example, $a = 2012_3 = 2 \cdot 3^3 + 0 \cdot 3^2 + 1 \cdot 3^1 + 2 \cdot 3^0 = 54 + 0 + 3 + 2 = 59.$
Similarly, $b = 201_3 = 2 \cdot 3^2 + 0 \cdot 3^1 + 1 \cdot 3^0 = 18 + 0 + 1 = 19.$
So, the product $ab$ is the same as the product $59 \cdot 19 = 1121.$
Now, I need to convert this product back to the base-three number system.
I can do this by repeatedly dividing by three and finding the remainders, which will be the digits of the base-three representation.
For example, $1121 \div 3 = 373$ with a remainder of $2,$ so the last digit is $2.$
Then, $373 \div 3 = 124$ with a remainder of $1,$ so the second-to-last digit is $1.$
Continuing this process, I get $124 \div 3 = 41$ with a remainder of $1,$ so the third-to-last digit is $1.$
Then, $41 \div 3 = 13$ with a remainder of $2,$ so the fourth-to-last digit is $2.$
Finally, $13 \div 3 = 4$ with a remainder of $1,$ so the fifth-to-last digit is $1,$ and $4 \div 3 = 1$ with a remainder of $1,$ so the sixth-to-last digit is $1.$
Therefore, the product $ab$ expressed in the base-three number system is $112112_3.$