### Instruction: What is the product (in base 10) of the first and last digits of the base-6 representation of $682_{10}$?### Response: To solve this problem, I need to convert 682 from base 10 to base 6, and then identify the first and last digits of the base-6 representation.
To convert from base 10 to base 6, I can use the repeated division method.
I start by dividing 682 by 6 and writing down the quotient and remainder.
682 divided by 6 is 113 with a remainder of 4.
This means that 4 is the rightmost digit of the base-6 representation of 682.
I repeat the process with the quotient 113.
113 divided by 6 is 18 with a remainder of 5.
This means that 5 is the next digit of the base-6 representation, to the left of 4.
I continue dividing by 6 until I reach a quotient of 0.
18 divided by 6 is 3 with a remainder of 0.
3 divided by 6 is 0 with a remainder of 3.
So the base-6 representation of 682 is 3054.
The first digit is 3 and the last digit is 4.
The product of these two digits is 3 times 4, which is 12.