Find the octal equivalent of $952_{10}$:

Question

Accepted Answer

The correct option is (1): $1670\_8$
To convert a decimal number like $952\_{10}$ to its octal (base-8) equivalent, we employ the repeated division-by-8 method. In this technique, the decimal number is successively divided by 8, and the remainder from each division becomes a digit in the octal result. Starting the process with 952, the first division ($952 \div 8$) yields a quotient of 119 and a remainder of $\mathbf{0}$. Dividing 119 by 8 gives a quotient of 14 with a remainder of $\mathbf{7}$. The next step divides 14 by 8, resulting in a quotient of 1 and a remainder of $\mathbf{6}$. Finally, dividing 1 by 8 gives a quotient of 0 and a remainder of $\mathbf{1}$. By collecting these remainders from the last one to the first (bottom-up), we correctly construct the octal equivalent, which is $\mathbf{1670\_8}$. This confirms that Option 1 is the mathematically accurate octal representation of the given decimal number.

The incorrect option is (2): $167\_8$
This option, $167\_8$, is incorrect because it represents a much smaller decimal value than 952. If we convert $167\_8$ back to decimal, it equals $(1 \times 8^2) + (6 \times 8^1) + (7 \times 8^0) = 64 + 48 + 7 = 119\_{10}$. This result shows that the conversion process was likely stopped too early, specifically after the second division, neglecting the remaining significant digits. The value $119\_{10}$ is merely the second quotient in the correct conversion process. For an accurate conversion, the division-by-8 process must always be continued until the final quotient is zero. Stopping prematurely omits the most significant digits of the octal number, leading to an error.

The incorrect option is (3): $1607\_8$
This option contains the correct digits but in an incorrect sequence, indicating a procedural error in recording the remainders. If we convert $1607\_8$ back to decimal, it yields a value of $(1 \times 8^3) + (6 \times 8^2) + (0 \times 8^1) + (7 \times 8^0) = 512 + 384 + 0 + 7 = 903\_{10}$. This calculation is close but is definitively not $952\_{10}$. The correct procedure requires reading the remainders strictly in reverse order—from the last remainder generated to the first. The sequence $1607$ suggests the final two remainders (7 and 0) were mistakenly swapped when being recorded in the final answer.

The incorrect option is (4): $1660\_8$
The octal number $1660\_8$ is incorrect because it reflects a mathematical error in one of the division steps. Converting this option back to decimal gives $(1 \times 8^3) + (6 \times 8^2) + (6 \times 8^1) + (0 \times 8^0) = 512 + 384 + 48 + 0 = 944\_{10}$. The correct decimal value is 952, showing a difference of 8. This error stems from miscalculating the remainder in the step where $119 \div 8$. The correct remainder is 7, but the digit '6' in this option suggests a remainder of 6 was recorded instead. In all number system conversions, meticulous arithmetic for every remainder is crucial for an accurate final result.