Convert the decimal number 71₁₀ to its binary (base-2) equivalent.

Question

Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context:Converting from decimal (base-10) to binary (base-2) is foundational in computer architecture and number systems. The standard approach repeatedly divides the decimal number by 2, tracking remainders, or uses powers of 2 decomposition. Given Data / Assumptions: Decimal input: 71₁₀. We want an unsigned binary representation. Binary weights: 1, 2, 4, 8, 16, 32, 64, ... Concept / Approach: Power-of-two decomposition: represent 71 as a sum of distinct powers of two. Alternatively, use successive division by 2 and read remainders in reverse. Step-by-Step Solution: Find largest power of 2 ≤ 71 → 64 = 2^6.Compute remainder: 71 - 64 = 7.Express 7 as 4 + 2 + 1 → 2^2 + 2^1 + 2^0.Set bits at positions 6, 2, 1, 0 → 1 000 111.Final binary: 1000111₂. Verification / Alternative check: Convert back: 164 + 032 + 016 + 08 + 14 + 12 + 1*1 = 64 + 0 + 0 + 0 + 4 + 2 + 1 = 71₁₀. Why Other Options Are Wrong: 110011₂: Equals 51₁₀, not 71. 1110011₂: Equals 115₁₀, not 71. 0110011₂: Equals 51₁₀ with a leading zero. None of the above: Incorrect because 1000111₂ is correct. Common Pitfalls: Reversing the order of remainders when using the division method. Dropping significant leading 1 in the highest place (2^6 here). Final Answer: 1000111₂

Convert the decimal number 71₁₀ to its binary (base-2) equivalent.

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