Octal arithmetic: Compute the product of 47₈ and 52₈ and express the result in octal.

Introduction / Context:
Working directly in octal reinforces base conversion fluency and place-value reasoning beyond decimal. While it is possible to multiply in octal directly, many learners find it reliable to convert to decimal (or binary), multiply, and then convert back to octal.

Given Data / Assumptions:

• Multiplicands: 47₈ and 52₈.
• We will verify using a decimal intermediate.
• No signs or fractions are involved.

Concept / Approach:
Strategy: convert each octal number to decimal, multiply, then convert the product back to octal. Formula reminders: for abc₈ = a8^2 + b8 + c, and to convert decimal back to octal, repeatedly divide by 8 and collect remainders.

Step-by-Step Solution:
Convert 47₈ to decimal: 48 + 7 = 32 + 7 = 39.Convert 52₈ to decimal: 58 + 2 = 40 + 2 = 42.Multiply in decimal: 39 * 42 = 1638.Convert 1638 to octal via successive division: 1638 / 8 → remainders yield 3 1 4 6 (from last to first).Thus, 1638₁₀ = 3146₈, so 47₈ * 52₈ = (3146)₈.

Verification / Alternative check:
Binary route: convert both to binary, multiply or use shift-add, then group bits in threes to return to octal. This also yields 3146₈.

Why Other Options Are Wrong:
(3144)₈, (4147)₈, (3184)₈: none equal 1638 in decimal when converted back; they correspond to different decimal values.None of the above: incorrect because (3146)₈ is correct.

Common Pitfalls:
Mixing base-10 and base-8 multiplication tables, or making remainder/order mistakes during the base conversion back to octal.

Final Answer:
(3146)8