What is the OCTAL equivalent of the HEXADECIMAL number (ADF)16 ?

Concept

Hexadecimal and octal are both powers of two — 16 = 24 and 8 = 23 — so binary is the common bridge between them. One hexadecimal digit is exactly four bits, and one octal digit is exactly three bits.

A conversion between the two therefore never needs decimal arithmetic: expand every digit of the source base into its fixed-width bit group, join the bits into one unbroken string, then re-partition that same string into groups of the target width, counting from the right and padding the leftmost group with zeros if it falls short.

Applying it here

1. Expand each hexadecimal digit of ADF into four bits: A = 1010, D = 1101, F = 1111.

2. Join them in source order into one binary string: 101011011111, twelve bits in all.

3. Twelve is a multiple of three, so no leading zeros are needed. Re-partition the same string from the right into three-bit groups: 101 | 011 | 011 | 111.

4. Convert each three-bit group to its octal digit: 101 = 5, 011 = 3, 011 = 3, 111 = 7.

5. Read the digits left to right: the octal equivalent is (5337)8.

The same bits, read two ways

Cross-check with positional weights

Both numerals must expand to the same decimal value. (ADF)16 = 10 × 162 + 13 × 16 + 15 = 2560 + 208 + 15 = 2783, and (5337)8 = 5 × 83 + 3 × 82 + 3 × 8 + 7 = 2560 + 192 + 24 + 7 = 2783. The two agree, so the regrouping is sound.

Where this method usually goes wrong

Two slips account for most errors: re-partitioning the bit string from the left instead of the right, and dropping a leading zero while expanding a hexadecimal digit (writing A as 101 instead of 1010). Either one shifts every group after it and changes the octal digits that come out.