Test for divisibility of numbers by 9 and 11
(g) Divisibility by 9
If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.
Example: Test of divisibility by 9
(i) 108
1 + 0 + 8 = 9
Here, the sum of the digits of 108 is equal to 9 and is divisible by 9, thus the number 108 is divisible by 9.
108 ÷ 9 = 12
(ii) 1269
1 + 2 + 6 + 9 = 18
Here, the sum of the digits of 1269 is equal to 18 and is divisible by 9, thus the number 1269 is divisible by 9.
1269 ÷ 9 = 141
(iii) 783
7 + 8 + 3 = 18
Here, the sum of the digits of 783 is equal to 18 and is divisible by 9, thus the number 783 is divisible by 9.
783 ÷ 9 = 87
(h) Divisibility of a number by 11
For a number, if the difference between the sum of digits at odd place and sum of digits at even place counting from the right is either zero (0) or divisible by 11, then the number is divisible by 11.
Example: (1) Test of divisibility of a number by 11 having difference between sum of digits at odd place and sum of digits at even place counting from right is equal to zero (0)
(i) 132:
In 132, the sum of digits of numbers at odd places counting from right
= 2 + 1 = 3
In 132, the sum of digits of numbers at even places counting from right
= 3
And difference between sum of digits at odd place and at even place counting from right for 132
= 3 – 3 = 0
Since, the difference between sum of digits at odd place and at even place counting from right for 132 is equal to 0
Thus, 132 is divisible by 11
132 ÷ 11 = 12
(ii) 154:
In 132, the sum of digits of numbers at odd places counting from right
= 4 + 1 = 5
In 154, the sum of digits of numbers at even places counting from right
= 5
And difference between sum of digits at odd place and at even place counting from right for 154
= 5 – 5 = 0
Since, the difference between sum of digits at odd place and at even place counting from right for 154 is equal to 0
Thus, 154 is divisible by 11
154 ÷ 11 = 14
(iii) 275:
In 275, the sum of digits of numbers at odd places counting from right
= 5 + 2 = 7
In 275, the sum of digits of numbers at even places counting from right
= 7
And difference between sum of digits at odd place and at even place counting from right for 275
= 7 – 7 = 0
Since, the difference between sum of digits at odd place and at even place counting from right for 275 is equal to 0
Thus, 275 is divisible by 11
275 ÷ 11 = 25
(iv) 2596:
In 2596, the sum of digits of numbers at odd places counting from right
= 6 + 5 = 11
In 2596, the sum of digits of numbers at even places counting from right
= 9 + 2 = 11
And difference between sum of digits at odd place and at even place counting from right for 2596
= 11 – 11 = 0
Since, the difference between sum of digits at odd place and at even place counting from right for 2596 is equal to 0
Thus, 2596 is divisible by 11
2596 ÷ 11 = 236
Example: (2) Test of divisibility of a number by 11 having difference between sum of digits at odd place and sum of digits at even place counting from right is equal to eleven (11)
(i) 715:
In 715, the sum of digits of numbers at odd places counting from right
= 5 + 7 = 12
In 715, the sum of digits of numbers at even places counting from right
= 1
And difference between sum of digits at odd place and at even place counting from right for 715
= 12 – 1 = 11
Since, the difference between sum of digits at odd place and at even place counting from right for 715 is equal to 11, thus, 715 is divisible by 11
715 ÷ 11 = 65
(ii) 1705:
In 1705, the sum of digits of numbers at odd places counting from right
= 5 + 7 = 12
In 715, the sum of digits of numbers at even places counting from right
= 0 + 1 = 1
And difference between sum of digits at odd place and at even place counting from right for 1705
= 12 – 1 = 11
Since, the difference between sum of digits at odd place and at even place counting from right for 1705 is equal to 11, thus, 1705 is divisible by 11
1705 ÷ 11 = 155
(iii) 7194:
In 7194, the sum of digits of numbers at odd places counting from right
= 4 + 1 = 5
In 7194, the sum of digits of numbers at even places counting from right
= 9 + 7 = 16
And difference between sum of digits at odd place and at even place counting from right for 7194
= 16 – 5 = 11
Since, the difference between sum of digits at odd place and at even place counting from right for 7194 is equal to 11, thus, 7194 is divisible by 11
7194 ÷ 11 = 654