Which one of the given options is the correct octal representation of the below given...

To convert decimal 519 to octal, repeatedly divide by 8 and collect remainders. 519 ÷ 8 = 64 rem 7, 64 ÷ 8 = 8 rem 0, 8 ÷ 8 = 1 rem 0, 1 ÷ 8 = 0 rem 1. Reading remainders bottom-to-top gives (1007)₈. Wait—let me recalculate: 519 ÷ 8 = 64 rem 7, 64 ÷ 8 = 8 rem 0, 8 ÷ 8 = 1 rem 0, 1 ÷ 8 = 0 rem 1, which gives 1007₈. Verification: 1×8³ + 0×8² + 0×8¹ + 7×8⁰ = 512 + 0 + 0 + 7 = 519 ✓. However, checking option C (1154): 1×8³ + 1×8² + 5×8¹ + 4×8⁰ = 512 + 64 + 40 + 4 = 620 ≠ 519. Re-examining: Actually 519 = 512 + 7 = 1×8³ + 0×8² + 0×8¹ + 7×8⁰ = (1007)₈. But let me verify option C differently. The correct answer should be (1007)₈, making option B correct.

Step-by-step Derivation:
Convert (519)₁₀ to octal using successive division by 8:

519 ÷ 8 = 64 remainder 7
64 ÷ 8 = 8 remainder 0
8 ÷ 8 = 1 remainder 0
1 ÷ 8 = 0 remainder 1

Reading remainders from bottom to top: (1007)₈

Verification: 1×8³ + 0×8² + 0×8¹ + 7×8⁰ = 512 + 0 + 0 + 7 = 519 ✓

Checking all options:
A) (256)₈ = 2×64 + 5×8 + 6 = 128 + 40 + 6 = 174 ✗
B) (1007)₈ = 1×512 + 0×64 + 0×8 + 7 = 519 ✓
C) (1154)₈ = 1×512 + 1×64 + 5×8 + 4 = 512 + 64 + 40 + 4 = 620 ✗
D) (1589)₈ = Invalid (digit 8 and 9 do not exist in octal) ✗