On dividing $(101011)_2$ by $(11)_2$ the quotient and remainder respectively are:

Question

Accepted Answer

The correct option is (1):
The solution requires performing binary long division, which is the standard method for dividing binary numbers, much like long division in decimal. The dividend is $(101011)\_2$, and the divisor is $(11)\_2$. In decimal terms, this is $43 \div 3$. We start the process by attempting to divide the first few bits of the dividend by the divisor. We find that $101\_2 \div 11\_2$ is $1$ with a remainder of $10\_2$. We continue the process, bringing down the subsequent bits and performing subtractions. The full division yields a quotient of $(1110)\_2$ and a final, non-zero remainder of $(1)\_2$. This can be confirmed by converting the result back to decimal, which is $14 \times 3 + 1 = 43$, thus verifying the correctness of the quotient and remainder.

The incorrect option is (2):
Option (2) suggests a quotient of $(1101)*2$ and a remainder of $(0)*2$. This is incorrect because the binary long division process clearly shows that the division is not exact and must result in a remainder of $(1)*2$. Furthermore, a quotient of $(1101)*2$ is $13*{10}$. Checking the division rule, $13 \times 3 + 0 = 39$. Since the original dividend is $43*{10}$, and $39 \neq 43$, the quotient is too small and the remainder is incorrect. This result would be the output of dividing $39*{10}$ by $3*{10}$, not $43\_{10}$.

The incorrect option is (3):
Option (3) suggests the correct quotient of $(1110)\_2$ but an incorrect remainder of $(0)*2$. The remainder must be exactly $(1)*2$ based on the final subtraction step in the binary long division. The proposed quotient $(1110)*2$ is $14*{10}$. If we verify this option using the division rule: $14 \times 3 + 0 = 42$. This value of $42*{10}$ does not equal the dividend of $43*{10}$. A remainder of $0$ is mathematically impossible when dividing an odd number, $43$, by an odd number, $3$, unless the result is an integer, which $43/3$ is not.

The incorrect option is (4):
Option (4) suggests a quotient of $(1011)*2$ and a remainder of $(1)*2$. Although the remainder is correct, the quotient is incorrect. The binary value $(1011)*2$ is $11*{10}$ in decimal. If we check this result: $11 \times 3 + 1 = 33 + 1 = 34$. This resulting value of $34*{10}$ is significantly less than the original dividend of $43*{10}$. An error in performing one of the intermediate subtractions or placements of a quotient bit during the long division would lead to an incorrectly low quotient like this.