Divisible by 9
Divisible by 9 is discussed below:
Consider the multiples of 9: 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, .......
|
Number |
Sum of the Digits |
|
18 54 72 |
1 + 8 = 9 5 + 4 = 9 7 + 2 = 9 |
In each case it can be observed that the sum of the digits is divisible by 9.
A number is divisible by 9, if the sum is a multiple of 9 or if the sum of its digits is divisible by 9.
1. Consider the following numbers which are divisible by 9, using the test of divisibility by 9:
99, 198, 171, 9990, 3411.
(i) 99
Sum of the digits of 99 = 9 + 9 = 18, which is divisible by 9.
Hence, 99 is divisible by 9.
(ii) 198
Sum of the digits of 198 = 1 + 9 + 8 = 18, which is divisible by 9.
Hence, 198 is divisible by 9.
(iii) 171
Sum of the digits of 171 = 1 + 7 + 1 = 9, which is divisible by 9.
Hence, 171 is divisible by 9.
(iv) 9990
Sum of the digits of 9990 = 9 + 9 + 9+ 0 = 27, which is divisible by 9.
Hence, 9990 is divisible by 9.
(v) 3411
Sum of the digits of 3411 = 3 + 4 + 1+ 1 = 9, which is divisible by 9.
Hence, 3411 is divisible by 9.
2. Consider the following numbers which are not divisible by 9, using the rules of divisibility by 9:
73, 237, 394, 1277, 1379.
Note: A number is divisible by 9 if the sum of its digits is divisible by 9.
(i) 73
Sum of the digits of 73 = 7 + 3 = 10, which is not divisible by 9.
Hence, 73 is not divisible by 9.
(ii) 237
Sum of the digits of 237 = 2 + 3 + 7 = 12, which is not divisible by 9.
Hence, 237 is not divisible by 9.
(iii) 394
Sum of the digits of 394 = 3 + 9 + 4 = 16, which is not divisible by 9.
Hence, 394 is not divisible by 9.
(iv) 1277
Sum of the digits of 1277 = 1 + 2 + 7 + 7 = 17, which is not divisible by 9.
Hence, 1277 is not divisible by 9.
(v) 1379
Sum of the digits of 1379 = 1 + 3 + 7 + 9 = 20, which is not divisible by 9.
Hence, 1379 is not divisible by 9.
3. Is 5328 divisible by 9?
Solution:
Sum of its digits = 5 + 3 + 2 + 8 = 18, divisible by 9.
Hence 5328 is divisible by 9.
We can verify that 5328 is divisible by 9 by actual division.
4. Which of the following numbers is not divisible by 9?
(i) 432
(ii) 5697
(iii) 7173
(iv) 1029
Solution:
(i) 432
Sum of the digits = 4 + 3 + 2 = 9, which is divisible by 9.
Thus 432 is divisible by 9.
(ii) 5697
Sum of the digits = 5 + 6 + 7 + 9 = 27, which is divisible by 9.
Thus 5697 is divisible by 9.
(iii) 7173
Sum of the digits = 7 + 1 + 7 + 3 = 18, which is divisible by 9.
Thus 7173 is divisible by 9.
(iv) 1029
Sum of the digits = 1 + 0 + 2 + 9 = 12, which is not divisible by 9.
Thus 7173 is not divisible by 9.
Answer: Therefore, the option (iv) is correct.
5. In each of the following numbers, replace * by a digit to make the number divisible by 9:
(i) 59 * 49; (ii) 21 * 664
Solution:
(i) 59 * 49
The sum of the digits in the number 59 * 49 = 5 + 9 + 4 + 9 = 27, which is divisible by 9.
Hence, 0 replaces * so as to make the number 59049 divisible by 9.
(ii) 21 * 664
The sum of the digits in the number 21 * 664 = 2 + 1 + 6 + 6 + 4 = 19, which is not multiples of 9.
Multiple of 9 after 19 is 27
So, 27 - 19 = 8 replaces * so as to make the number 218664 divisible by 9.
Answer: (i) 0
(ii) 8
Worksheet on Divisible by 9:
1. Choose the numbers which are divisible by 9.
13 27 128 65 730363 900 1375
Answer: 27, 900
2. Which of the following numbers are divisible by 9?
(i) 855
(ii) 97
(iii) 4830
(iv) 9189
(v) 8469
Answer:
2. (i) 855
(iv) 9189
(v) 8469
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