(12178) is equivalent to:

Concept

Octal (base 8) and hexadecimal (base 16) are both powers of 2: 8 = 23 and 16 = 24. So each octal digit maps to exactly 3 binary bits, and each hex digit to exactly 4 binary bits. The clean route between the two bases is therefore through binary: expand to bits using 3 per octal digit, then re-group those same bits into 4s for hex.

Application

1. Expand each octal digit of 12178 into 3 bits: 1 = 001, 2 = 010, 1 = 001, 7 = 111.

2. Concatenate the bits in order: 001 010 001 111, i.e. the binary string 001010001111.

3. Re-group the same 12 bits into 4-bit chunks from the right: 0010 | 1000 | 1111.

4. Read each chunk as a hex digit: 0010 = 2, 1000 = 8, 1111 = F.

5. Assemble left to right: 2, 8, F, giving (28F16), written with a leading zero as (028F16).

Cross-check

Convert via decimal as an independent check: 12178 = 1×83 + 2×82 + 1×81 + 7×80 = 512 + 128 + 8 + 7 = 65510. And 655 = 2×162 + 8×161 + 15×160 = 2,8,F = 28F16. Both routes agree.