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 185472 1 + 8 = 95 + 4 = 97 + 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.

(ii) 8

Worksheet on Divisible by 9:

1. Choose the numbers which are divisible by 9.

13          27          128          65          730363          900          1375

2. Which of the following numbers are divisible by 9?

(i) 855

(ii) 97

(iii) 4830

(iv) 9189

(v) 8469

2. (i) 855

(iv) 9189

(v) 8469

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Properties of Divisibility.

Divisible by 2.

Divisible by 3.

Divisible by 4.

Divisible by 5.

Divisible by 6.

Divisible by 7.

Divisible by 8.

Divisible by 9.

Divisible by 10.

Problems on Divisibility Rules

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