Convert 112$_6$ to a number in base three

1. Analyze the question

We are asked to convert a number given in base six ($112_6$) into base three. Since there is no direct way to convert between base six and base three, we must use Base 10 as an intermediary.

2. Explain the concept

• Step 1: Convert the number from base six to base ten by multiplying each digit by its corresponding power of 6.
• Step 2: Convert the resulting base ten number to base three by using the method of successive division (dividing by 3 and keeping track of the remainders).

3. Show the solution

Step 1: Convert $112_6$ to base 10 $112_6 = (1 \times 6^2) + (1 \times 6^1) + (2 \times 6^0)$ $112_6 = (1 \times 36) + (1 \times 6) + (2 \times 1)$ $112_6 = 36 + 6 + 2$ $112_6 = 44_{10}$

Step 2: Convert $44_{10}$ to base 3 We divide 44 by 3 repeatedly and record the remainders:

• $44 \div 3 = 14$ remainder 2
• $14 \div 3 = 4$ remainder 2
• $4 \div 3 = 1$ remainder 1
• $1 \div 3 = 0$ remainder 1

Reading the remainders from the bottom to the top, we get 1122.

4. Identify the answer

Comparing our result ($1122_3$) to the given options, it matches option D.

5. Tip

When converting from base 10 to any other base, always remember to read your remainders from the last one obtained (bottom) to the first one obtained (top). A common mistake students make is reading them from top to bottom, which will give you the wrong answer!