What is representation of $(539)_{10}$ in hexadecimal ?

Question

Accepted Answer

The correct option is (4):
The correct hexadecimal representation is found by repeatedly dividing the decimal number, $539\_{10}$, by 16 and collecting the remainders. For the first division, $539 \div 16 = 33$ with a remainder of 11. In the hexadecimal system, the decimal value 11 is represented by the digit 'B', which becomes the least significant digit. Next, we divide the quotient 33 by 16, yielding $33 \div 16 = 2$ with a remainder of 1. Finally, dividing the quotient 2 by 16 gives $2 \div 16 = 0$ with a remainder of 2. Reading the remainders from bottom (most significant) to top (least significant) gives the hexadecimal result, which is $\text{21B}$.

The incorrect option is (1):
Option ABE is incorrect. To verify this, we convert $\text{ABE}*{16}$ back to decimal using positional weightings: $(\text{A} \times 16^2) + (\text{B} \times 16^1) + (\text{E} \times 16^0)$. By substituting the decimal values for the letters (A=10, B=11, E=14), the calculation is $(10 \times 256) + (11 \times 16) + (14 \times 1)$. This results in $2560 + 176 + 14$, which sums up to $2750*{10}$. Since $2750$ is not equal to $539$, $\text{ABE}$ is a significantly larger number and is therefore incorrect.

The incorrect option is (2):
Option CDB is incorrect. Similar to the previous option, a three-digit hexadecimal number starting with C (decimal 12) suggests a much larger magnitude than 539. Converting $\text{CDB}*{16}$ to decimal gives us $(\text{C} \times 16^2) + (\text{D} \times 16^1) + (\text{B} \times 16^0)$. This evaluates to $(12 \times 256) + (13 \times 16) + (11 \times 1)$. The total decimal value is $3072 + 208 + 11 = 3291*{10}$. This result is substantially different from the required value of $539\_{10}$.

The incorrect option is (3):
Option DES is incorrect primarily because it contains an invalid digit. The hexadecimal (base-16) number system uses only 16 unique symbols: the digits 0 through 9, and the letters A, B, C, D, E, and F to represent the decimal values 10 through 15. The letter 'S' is not a symbol defined within the set of hexadecimal digits. Therefore, any string that includes the letter 'S' is not a valid representation of a number in the hexadecimal system.