Convert the unsigned octal number 67675 to binary.

Question

BlackTom AI · Accepted Answer

To begin, I will restate the task: convert the unsigned octal number 67675 to its binary representation, using the standard 3-bit mapping for each octal digit.
First, recall the direct octal-to-binary conversion rule: each octal digit corresponds to exactly three binary bits. This means we can convert digit-by-digit without performing more complex arithmetic.
Looking at the digits of 67675 in octal, we have the sequence 6, 7, 6, 7, 5.
- The octal digit 6 converts to binary as 110.
- The octal digit 7 converts to binary as 111.
- The next octal digit 6 again converts to binary as 110.
- The following 7 converts to binary as 111.
- Finally, the octal digit 5 converts to binary as 101.
Assembling these 3-bit groups in order yields: 110 111 110 111 101.
Now, evaluate the provided options:
- Option A: 110 111 110 111 101 — This matches the exact concatenation of the per-digit binary equivalents, so it correctly represents 67675 in binary.
- Option B: 111 011 110 111 101 — Here the first group is 111 instead of 110, which would imply a different first octal digit, so this option misrepresents the conversion.
- Option C: 111 010 110 101 111 — This sequence disrupts the grouping pattern and does not align with any standard per-digit conversion for 6, 7, 6, 7, 5, indicating an incorrect mapping.
- Option D: 110 010 110 101 011 — This option alters several groups compared to the correct mapping, showing improper conversions for at least some digits.
- Option E: None of these — Since the first option exactly matches the correct per-digit binary equivalents, this choice is contradicted by the valid result.
In summary, the correct approach is to translate each octal digit independently into its 3-bit binary form and then concatenate the results. Any deviation in the bit groups indicates a misapplication of the octal-to-binary mapping, which is why the other options fail to represent 67675 accurately.

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