If (P)8 represents the number P in base -8 number system, then (124)8 + (165)8 =

Concept

In a positional base-b numeral, each digit is multiplied by a power of b determined by its place, starting with b⁰ at the right.

Numbers in the same base can be added by converting their place values or by carrying within that base. In base 8, the permitted digits are 0 through 7.

Application

1. Expand the first numeral: (124)₈ = 1 × 8² + 2 × 8 + 4 = 84₁₀.

2. Expand the second numeral: (165)₈ = 1 × 8² + 6 × 8 + 5 = 117₁₀.

3. Add the decimal values: 84 + 117 = 201.

4. Convert 201 back to base 8: 201 = 25 × 8 + 1; 25 = 3 × 8 + 1; 3 = 0 × 8 + 3. Reading the remainders upward gives (311)₈.

Cross-check

Octal column addition gives 4 + 5 = 9 = (11)₈, so write 1 and carry 1. Then 2 + 6 + 1 = 9 = (11)₈, so again write 1 and carry 1. Finally, 1 + 1 + 1 = 3, confirming (311)₈.

Therefore, (124)₈ + (165)₈ = (311)₈.