What will be the binary equivalent of (562)8?

Concept: Octal is base 8 and 8 = 23, so one octal digit is worth exactly three binary bits. Octal-to-binary is therefore a direct digit-by-digit substitution: replace every octal digit by its own 3-bit binary code and keep the digits in the same left-to-right order. Nothing is added and nothing is carried from one digit to the next.

Application — converting (562)8:

1. Write the octal digits of (562)8 separately: 5, 6 and 2.

2. 5 = 4 + 1, so in three bits it is 101.

3. 6 = 4 + 2, so in three bits it is 110.

4. 2 = 2, so in three bits it is 010.

5. Join the three groups in the same order — 101 110 010 — which gives (101110010)2.

Cross-check — convert both sides to decimal:

• (562)8 = 5 × 82 + 6 × 81 + 2 × 80 = 320 + 48 + 2 = 370.

• (101110010)2 = 256 + 64 + 32 + 16 + 2 = 370.

• Both sides give 370, so the three-bit grouping is right. Note that 110 stands for 6 while 111 stands for 7 — swapping these two codes is the most common slip in this conversion.

Result: (562)8 = (101110010)2.