Select the binary representation of 5116.

Question

BlackTom AI · Accepted Answer

Binary representation is built from powers of two. 51 in decimal equals 32 + 16 + 2 + 1, i.e. 2^5 + 2^4 + 2^1 + 2^0, so its binary digits at positions 5..0 are 1,1,0,0,1,1, giving 110011₂.

Relevant concepts:
- Binary (base 2) digits represent coefficients of powers of two: bit positions from right to left are 2^0, 2^1, 2^2, etc.
- Converting decimal to binary can be done by repeated division by 2, recording remainders (least significant bit first).

Step-by-step division method (showing all intermediate steps):
- 51 ÷ 2 = 25 remainder 1  (LSB)
- 25 ÷ 2 = 12 remainder 1
- 12 ÷ 2 = 6  remainder 0
- 6  ÷ 2 = 3  remainder 0
- 3  ÷ 2 = 1  remainder 1
- 1  ÷ 2 = 0  remainder 1  (stop when quotient 0)

Reading remainders from last to first: 1 1 0 0 1 1 → (110011)₂.

Why this is correct:
- Evaluate (110011)₂ = 1·2^5 + 1·2^4 + 0·2^3 + 0·2^2 + 1·2^1 + 1·2^0 = 32 + 16 + 0 + 0 + 2 + 1 = 51, matching the decimal value.

Why the other choices are incorrect (convert each to decimal):
- (1010001)₂ = 1·2^6 + 0·2^5 + 1·2^4 + 0·2^3 + 0·2^2 + 0·2^1 + 1·2^0 = 64 + 16 + 1 = 81 ≠ 51.
- (110111)₂ = 1·2^5 + 1·2^4 + 0·2^3 + 1·2^2 + 1·2^1 + 1·2^0 = 32 + 16 + 0 + 4 + 2 + 1 = 55 ≠ 51.
- (111011)₂ = 1·2^5 + 1·2^4 + 1·2^3 + 0·2^2 + 1·2^1 + 1·2^0 = 32 + 16 + 8 + 0 + 2 + 1 = 59 ≠ 51.

Conclusion: The correct binary representation is (110011)₂, which matches the provided answer.

Select the binary representation of 5116.单项选择题

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