Which of the following is the correct octal representation of the hexadecimal number 1A3?

Explanation:

Hexadecimal to Octal Conversion

Definition: Hexadecimal and octal are both positional numeral systems, widely used in computing and digital electronics. Hexadecimal (base-16) uses sixteen symbols, 0-9 and A-F, to represent values, while octal (base-8) uses eight symbols, 0-7.

To convert a hexadecimal number to its octal equivalent, it is often easiest to first convert it to binary (base-2), and then from binary to octal. This method works because both octal and hexadecimal are powers of 2 (octal is 2³ and hexadecimal is 2⁴).

Step-by-Step Solution:

Given hexadecimal number: 1A3

1. Convert Hexadecimal to Binary:

Each hexadecimal digit can be represented by a 4-bit binary number:

• 1 (Hex) = 0001 (Binary)
• A (Hex) = 1010 (Binary)
• 3 (Hex) = 0011 (Binary)

So, the hexadecimal number 1A3 can be written in binary as: 0001 1010 0011

2. Group Binary Digits into Sets of Three:

Since octal is base-8 and each octal digit corresponds to 3 binary digits, we group the binary digits in sets of three, starting from the right:

• 0001 1010 0011 (Binary)
• 000 110 100 011 (Binary, grouped in sets of three)

3. Convert Binary Sets to Octal:

Each group of three binary digits can be converted directly to its octal equivalent:

• 000 (Binary) = 0 (Octal)
• 110 (Binary) = 6 (Octal)
• 100 (Binary) = 4 (Octal)
• 011 (Binary) = 3 (Octal)

So, the binary number 000 110 100 011 can be written in octal as: 0643

Hence, the correct octal representation of the hexadecimal number 1A3 is 643.

Important Information:

To analyze other options, let's convert the hexadecimal number 1A3 using the same method:

• Option 1: 346
  • 346 (Octal) in binary: 011 100 110
  • Groups: 011 100 110 (Binary) = 3 4 6 (Octal)
  • Binary to Hexadecimal: 011100110 (Binary) = 1C6 (Hex)
  • 1C6 ≠ 1A3
• Option 2: 124
  • 124 (Octal) in binary: 001 010 100
  • Groups: 001 010 100 (Binary) = 1 2 4 (Octal)
  • Binary to Hexadecimal: 001010100 (Binary) = 54 (Hex)
  • 54 ≠ 1A3
• Option 3: 634
  • 634 (Octal) in binary: 110 011 100
  • Groups: 110 011 100 (Binary) = 6 3 4 (Octal)
  • Binary to Hexadecimal: 110011100 (Binary) = 19C (Hex)
  • 19C ≠ 1A3

Therefore, the correct option is confirmed as option 4: 643.