Convert decimal 54807 to binary

Question

BlackTom AI · Accepted Answer

Repeated division by 2 (or equivalently summing powers of two) converts 54807 into binary. Key concepts: each binary digit (bit) represents a power of two; repeatedly dividing the decimal number by 2 and recording remainders yields bits from least significant (LSB) to most significant (MSB).

Step-by-step division and remainders (quotient ÷ 2 = new quotient, remainder):
1) 54807 ÷ 2 = 27403 R1
2) 27403 ÷ 2 = 13701 R1
3) 13701 ÷ 2 = 6850  R1
4) 6850 ÷ 2  = 3425  R0
5) 3425 ÷ 2  = 1712  R1
6) 1712 ÷ 2  = 856   R0
7) 856 ÷ 2   = 428   R0
8) 428 ÷ 2   = 214   R0
9) 214 ÷ 2   = 107   R0
10) 107 ÷ 2  = 53    R1
11) 53 ÷ 2   = 26    R1
12) 26 ÷ 2   = 13    R0
13) 13 ÷ 2   = 6     R1
14) 6 ÷ 2    = 3     R0
15) 3 ÷ 2    = 1     R1
16) 1 ÷ 2    = 0     R1  (MSB)

Reading the remainders from last (step 16) to first (step 1) gives the binary string:
1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 1
So binary = 1101011000010111.

Verify by summing powers of two for positions with 1s (indexing bits 15 down to 0):
2^15(32768) + 2^14(16384) + 2^12(4096) + 2^10(1024) + 2^9(512) + 2^4(16) + 2^2(4) + 2^1(2) + 2^0(1)
= 32768 + 16384 + 4096 + 1024 + 512 + 16 + 4 + 2 + 1
= 54807

Why the correct option is correct: the option 1101011000010111 matches the bits derived by division and its weighted sum equals 54807 exactly.

Why the other choices are incorrect: each other binary string differs in one or more bit positions (especially the lower bits), changing the sum of corresponding powers of two and thus representing a different decimal value. For example, 1101011000011110 has lower bits 1110 instead of 0111 and would sum to a different decimal. Therefore they do not equal 54807.

Conclusion: the correct binary representation of decimal 54807 is 1101011000010111, which matches the provided answer.

Convert decimal 54807 to binary单项选择题

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