1. The rule for divisibility by 9 is similar to divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.
For example, with 2871, you just add \(2 + 8 + 7 + 1 = 18\). Since \(18\) is divisible by \(9\), the whole number is too.
2. Divisibility Rule for 11
A number is divisible by 11 if the difference between the sum of digits at odd positions and the sum of digits at even positions is either 0 or a multiple of 11.
Steps to Check Divisibility
- Write the digits of the number.
- Find:
- Sum of digits in odd positions
- Sum of digits in even positions
- Find the difference between these two sums.
- If the result is:
- 0, or
- multiple of 11 (±11, ±22, …)Then the number is divisible by 11.
Example 1
Check 121
- Odd position digits: \(1 + 1 = 2\)
- Even position digit: \(2\)
\[
2 – 2 = 0
\]
Divisible by 11
Example 2
Check 1331
- Odd position digits: \(1 + 3 = 4\)
- Even position digits: \(3 + 1 = 4\)
\[
4 – 4 = 0
\]
Divisible by 11
Example 3
Check 12345
- Odd position digits: \(1 + 3 + 5 = 9\)
- Even position digits: \(2 + 4 = 6\)
\[
9 – 6 = 3
\]
Not divisible by 11
Shortcut Trick
Example: 918273
\[
9 – 1 + 8 – 2 + 7 – 3 = 18
\]
Since 18 is not a multiple of 11, the number is not divisible by 11.