Add 54 $_{eight}$ and 67$_{eight}$ giving your answers in base eight

Analysis

To solve this problem, we need to perform addition in base eight (octal). In base eight, the digits allowed are 0, 1, 2, 3, 4, 5, 6, and 7. Whenever the sum of two digits is 8 or more, we must "carry over" to the next column, just as we do in base ten, but using 8 as the threshold.

Concept: Addition in Base Eight

When adding in base eight:

1. Add the digits in each column starting from the right.
2. If the sum is less than 8, write it down.
3. If the sum is 8 or greater, divide the sum by 8. The remainder is written down, and the quotient is carried over to the next column.

Step-by-Step Solution

Step 1: Add the rightmost column (units column) $4 + 7 = 11$ (in base ten). Since 11 is greater than 8, we divide by 8: $11 \div 8 = 1$ remainder $3$. Write down 3 and carry over 1.

Step 2: Add the next column (eights column) $5 + 6 = 11$. Now, add the carry-over from the previous step: $11 + 1 = 12$ (in base ten). Since 12 is greater than 8, we divide by 8: $12 \div 8 = 1$ remainder $4$. Write down 4 and carry over 1.

Step 3: Add the final column There are no more digits to add, so we bring down the carried-over 1.

Result: Combining these, we get 143.

Conclusion

The sum of $54_8$ and $67_8$ is $143_8$. This matches option D.

Exam Tip

Always remember that in any base $n$, the maximum digit you can have is $n-1$. If you are adding in base eight and your sum is 8, you write 0 and carry 1. If your sum is 9, you write 1 and carry 1. Never write a digit 8 or higher in your final answer for base eight!