What is the product (in base 10) of the first and last digits of the base-6 representation of $682_{10}$?
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. 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. 