Convert the decimal number 67₁₀ to its octal representation (showing the correctly base-tagged form).

Introduction / Context:
Base conversion is a staple skill in digital systems. Octal is convenient because each octal digit corresponds to three binary bits, providing a compact representation of binary values (especially in early microprocessor documentation and permissions in operating systems). Here we convert a base-10 integer to base 8.

Given Data / Assumptions:

• Decimal input n = 67.
• Target base = 8 (octal).
• Output expressed with an explicit base tag 8.

Concept / Approach:

Use repeated division by the target base and collect remainders. For octal, divide by 8; the remainders (0–7) form digits from least to most significant when read in reverse order of computation. This algorithm mirrors positional weighting in base systems.

Step-by-Step Solution:

Verification / Alternative check:

Decimal check: 18^2 + 08^1 + 3*8^0 = 64 + 0 + 3 = 67, confirming correctness.

Why Other Options Are Wrong:

1008 represents 64, not 67. 1098 exceeds digit range (digit 9 is invalid in octal). 998 is invalid (9 not allowed). 'None of the above' is wrong because 1038 is correct.

Common Pitfalls:

Forgetting to reverse the remainder order, or mistakenly using base 2/16 groupings instead of base 8 division.

Final Answer:

1038