Which of the following value is not equal to $(7214)_8$?

Question

Accepted Answer

The correct option is (2):
To determine which value is *not* equal to $(7214)*8$, we must convert the original octal number and all options to a common base, typically the decimal system (base 10), for a direct comparison. The octal number $(7214)*8$ is calculated by positional weighting as $(7 \times 8^3) + (2 \times 8^2) + (1 \times 8^1) + (4 \times 8^0)$, which equals $3584 + 128 + 8 + 4$, resulting in $3724*{10}$. Now, let us examine Option 2, the hexadecimal number $(EAC)*{16}$. In hexadecimal, $E$ represents $14$, $A$ represents $10$, and $C$ represents $12$. Converting this gives $(14 \times 16^2) + (10 \times 16^1) + (12 \times 16^0) = 3584 + 160 + 12$, which sums up to $3756\_{10}$. Since $3756\_{10}$ is not equal to $3724\_{10}$, Option 2 is the value that does not match the original octal number, making it the correct choice for this question.

The incorrect option is (1):
Option 1, the binary number $(111010001100)\_2$, is mathematically equal to the original octal number $(7214)\_8$. This equivalence is readily verifiable by the shortcut conversion method between octal and binary, where each octal digit corresponds to exactly three binary bits. Converting the octal digits $7, 2, 1, 4$ to their three-bit binary forms, we get $111, 010, 001, 100$ respectively. Concatenating these segments yields exactly the binary string $(111010001100)\_2$. Therefore, this value is equal to the target number, and thus it is an incorrect choice for the question asking for the *not equal* value.

The incorrect option is (3):
Option 3 is explicitly given as the decimal number $(3724)\_{10}$. As established in the primary calculation, the conversion of the octal number $(7214)\_8$ to its decimal base is precisely $3724$. Because this option is numerically identical to the value of the octal number, it is equal to the target value. Consequently, it must be ruled out as an answer, as it fails the condition of being unequal to the octal number.

The incorrect option is (4):
Option 4, the base 4 number $(322030)*4$, is also numerically equal to the original octal number. Converting this to decimal involves calculating $(3 \times 4^5) + (2 \times 4^4) + (2 \times 4^3) + (0 \times 4^2) + (3 \times 4^1) + (0 \times 4^0)$. This weighted sum evaluates to $3072 + 512 + 128 + 0 + 12 + 0$, which correctly totals $3724*{10}$. Since $(322030)\_4$ is mathematically equivalent to $(7214)\_8$, it is an incorrect option because it is equal to the octal number.