Which of the following value is not equal to $(6103)_8$?

Question

Accepted Answer

The correct option is (4):
To determine which value is not equal to $(6103)*8$, we first need to convert the octal number to its decimal equivalent, as decimal is the easiest base for direct comparison. The value of $(6103)*8$ in base 10 is calculated as: $6 \times 8^3 + 1 \times 8^2 + 0 \times 8^1 + 3 \times 8^0 = 6 \times 512 + 1 \times 64 + 0 + 3 = 3072 + 64 + 3 = 3139*{10}$. We must then convert the base-4 number in Option 4, $(30103)*4$, to decimal to see if it matches. The conversion is: $3 \times 4^4 + 0 \times 4^3 + 1 \times 4^2 + 0 \times 4^1 + 3 \times 4^0 = 3 \times 256 + 0 + 1 \times 16 + 0 + 3 = 768 + 16 + 3 = 787*{10}$. Since $787*{10}$ is not equal to $3139\_{10}$, this value is the one that is mathematically unequal to the original octal number, making it the correct answer to the question.

The incorrect option is (1):
Option 1, the binary number $(110001000011)\_2$, is in fact equal to $(6103)\_8$. A rapid method for this verification is the Binary-to-Octal conversion, which involves grouping the binary digits into sets of three starting from the right-most digit, since $2^3 = 8$. Grouping $(110001000011)\_2$ yields $110\ 001\ 000\ 011$. Converting each group to octal: $110\_2$ equals $6\_8$, $001\_2$ equals $1\_8$, $000\_2$ equals $0\_8$, and $011\_2$ equals $3\_8$. Concatenating these digits results in $(6103)\_8$, proving its equality to the given number.

The incorrect option is (2):
Option 2, the hexadecimal number $(C43)\_{16}$, also represents the same value as $(6103)*8$. To prove this, we convert the hexadecimal number to decimal. Remember that the hexadecimal digit 'C' corresponds to the decimal value 12. The decimal calculation is: $C \times 16^2 + 4 \times 16^1 + 3 \times 16^0 = 12 \times 256 + 4 \times 16 + 3 \times 1$. This arithmetic works out to $3072 + 64 + 3$, which totals $3139*{10}$. Since this decimal value exactly matches the decimal equivalent of the original octal number, this option is equal and therefore incorrect as the answer.

The incorrect option is (3):
Option 3, the decimal number $(3139)\_{10}$, is simply the decimal equivalent of the original octal number, which we established in the primary conversion step. The question requires you to identify the value that is *not* equal to $(6103)\_8$. Since we found that $6 \times 8^3 + 1 \times 8^2 + 0 \times 8^1 + 3 \times 8^0$ precisely equals $3139$, this option is definitively equivalent. Consequently, this value is a correct representation of the original number and cannot be the answer to the question asking for the unequal value.