What is the octal equivalent of (F3B1) hexadecimal number ?

Concept: A hexadecimal digit is a 4-bit binary pattern (since 16 = 24), while an octal digit is a 3-bit binary pattern (since 8 = 23). So to convert hex to octal, first expand every hex digit into its 4-bit binary form and concatenate all the bits into one binary string; then regroup that same string into 3-bit chunks counted from the right (padding the leftmost chunk with zeros if the total bit-length is not a multiple of 3), and read each 3-bit chunk as one octal digit.

Application — converting (F3B1) hexadecimal to octal:

1. Expand each hex digit into its 4-bit binary form: F = 1111, 3 = 0011, B = 1011, 1 = 0001.

2. Concatenate the four groups in order: 1111 0011 1011 0001, giving the 16-bit binary string 1111001110110001.

3. 16 is not a multiple of 3, so pad two leading zeros to make an 18-bit string, then regroup from the right into 3-bit chunks: 00 1111 0011 1011 0001 → 001 111 001 110 110 001.

4. Convert each 3-bit chunk to its octal digit: 001 = 1, 111 = 7, 001 = 1, 110 = 6, 110 = 6, 001 = 1.

5. Reading the digits in order gives the octal number 171661.

Cross-check — an independent route via decimal, instead of binary grouping:

1. Convert (F3B1) hexadecimal to decimal: 15 × 163 + 3 × 162 + 11 × 16 + 1 = 61440 + 768 + 176 + 1 = 62385.

2. Convert 62385 to octal by repeated division by 8, tracking each remainder: 62385÷8 = 7798 r1, 7798÷8 = 974 r6, 974÷8 = 121 r6, 121÷8 = 15 r1, 15÷8 = 1 r7, 1÷8 = 0 r1.

3. Reading the remainders from the last division to the first gives 1 7 1 6 6 1 = 171661 — the same result as the binary-grouping method.

Both independent methods agree: the octal equivalent of (F3B1) hexadecimal is 171661.