2.8 Divisibility by 9

Rule: Sum of digits divisible by 9. Reasoning: Identical logic to divisibility by 3, since 10 ≡ 1 (mod 9) as well (10 = 9+1).

Example 1: 4,86,729 → digit sum = 4+8+6+7+2+9 = 36, divisible by 9 → divisible by 9. Example 2: Find remainder when 71,364 divided by 9. Digit sum = 7+1+3+6+4 = 21 → 21÷9 leaves remainder 3. Number leaves remainder 3 when divided by 9.

Example 3: Is 6,58,932 divisible by 9? What is the remainder if not? Digit sum = 6+5+8+9+3+2 = 33. 33 ÷ 9 = 3 remainder 6. Since the digit sum is not a multiple of 9, 6,58,932 is not divisible by 9, and it leaves remainder 6.

Example 4: Find the remainder when 1,23,456 is divided by 9. Digit sum = 1+2+3+4+5+6=21. 21 ÷ 9 leaves remainder 3. Remainder = 3.