Calculate 243$_{six}$ – 243$_{five}$ expressing your answer in base 10

To solve this problem, we need to convert both numbers from their respective bases to base 10 and then find the difference.

1. Analyze the question

We are given two numbers in different bases:

• $243_{six}$
• $243_{five}$ We need to convert both to base 10 and subtract the second from the first.

2. Explain the concept

To convert a number from any base ($n$) to base 10, we multiply each digit by the base raised to the power of its position (starting from 0 on the far right).

The formula is: $d_n \times n^n + ... + d_1 \times n^1 + d_0 \times n^0$

3. Show the solution

Step 1: Convert $243_{six}$ to base 10 $243_{six} = (2 \times 6^2) + (4 \times 6^1) + (3 \times 6^0)$ $= (2 \times 36) + (4 \times 6) + (3 \times 1)$ $= 72 + 24 + 3$ $= 99_{ten}$

Step 2: Convert $243_{five}$ to base 10 $243_{five} = (2 \times 5^2) + (4 \times 5^1) + (3 \times 5^0)$ $= (2 \times 25) + (4 \times 5) + (3 \times 1)$ $= 50 + 20 + 3$ $= 73_{ten}$

Step 3: Subtract the results $99 - 73 = 26$

4. Identify the answer

Comparing our result ($26$) with the given options: A. 0 B. 1 C. 26 D. 46

The correct option is C.

5. Tip

When converting from any base to base 10, always remember to start your powers from the rightmost digit (the units place) starting at 0. A common mistake is starting from 1; always remember that $x^0 = 1$, not 0!