Binary of 678 decimal – UGC NET Question

To convert 678 to binary, we repeatedly divide by 2 and record the remainders, obtaining the pattern 1010100110 when written from most significant to least significant bit. Checking by expansion, 1010100110₂ equals 512 + 128 + 32 + 4 + 2 = 678. This confirms that 1010100110 is the correct binary representation. The other binary strings expand to different decimal values.

Option A:
1010100010₂ corresponds to 512 + 128 + 32 + 2 = 674. Since 674 is not equal to 678, this pattern is close but not correct. It illustrates how a single bit difference changes the decimal result.

Option B:
1010100110₂ expands as 1×512 + 0×256 + 1×128 + 0×64 + 1×32 + 0×16 + 0×8 + 1×4 + 1×2 + 0×1. Adding these values gives 512 + 128 + 32 + 4 + 2 = 678, exactly matching the required decimal. This confirms that this option is the correct binary conversion.

Option C:
1010110110₂ represents 512 + 128 + 32 + 16 + 8 + 4 + 2 = 702. As the decimal value is larger than 678, this cannot be the correct representation. It contains more 1 bits in higher positions, increasing its value.

Option D:
1010010110₂ equals 512 + 128 + 4 + 2 = 646. Because 646 does not equal 678, this option is not correct. It has fewer high-order bits set, so the decimal value is smaller than required.