A number is divisible by 9 and 100 both by which other number will that number be always divisible

Table of Contents Show

Divisible By
Factors Can Be Useful
Another Rule For 11
Example: 14641
What is the Divisibility Rule of 9?
Divisibility Rule of 9 for Large Numbers
Divisibility by 3 and 9
Divisibility Rule of 9 and 11
FAQs on Divisibility Rule of 9
What do the Divisibility Rules for 9 and 3 have in Common?
Using the Divisibility Rule of 9, Check if 1450 is Divisible by 9?
How do you know if a Big Number is Divisible by 9?
Using the Divisibility Rule of 9, Check if 8955 is Divisible by 9?
What is the Divisibility Rule of 3 and 9?
Test Whether 9846 is Divisible by 9.

Easily test if one number can be exactly divided by another

Divisible By

"Divisible By" means "when you divide one number by another the result is a whole number"

14 is divisible by 7, because 14 ÷ 7 = 2 exactly

15 is not divisible by 7, because 15 ÷ 7 = 2 17 (the result is not a whole number)

0 is divisible by 7, because 0 ÷ 7 = 0 exactly (0 is a whole number)

"Divisible by" and "can be exactly divided by" mean the same thing

These rules let you test if one number is divisible by another, without having to do too much calculation!

We could try dividing 723 by 3

Or use the "3" rule: 7+2+3=12, and 12 ÷ 3 = 4 exactly Yes

Note: Zero is divisible by any number (except by itself), so gets a "yes" to all these tests.

Any integer (not a fraction) is divisible by 1

The last digit is even (0,2,4,6,8)

128 Yes

129 No

The sum of the digits is divisible by 3

381 (3+8+1=12, and 12÷3 = 4) Yes

217 (2+1+7=10, and 10÷3 = 3 1/3) No

This rule can be repeated when needed:

99996 (9+9+9+9+6 = 42, then 4+2=6) Yes

The last 2 digits are divisible by 4

1312 is (12÷4=3) Yes

7019 is not (19÷4=4 3/4) No

We can also subtract 20 as many times as we want before checking:

68: subtract 3 lots of 20 and we get 8 Yes

102: subtract 5 lots of 20 and we get 2 No

Another method is to halve the number twice and see if the result is still a whole number:

124/2 = 62, 62/2 = 31, and 31 is a whole number. Yes

30/2 = 15, 15/2 = 7.5 which is not a whole number. No

The last digit is 0 or 5

175 Yes

809 No

Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)

114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes

308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No

Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7. (We can apply this rule to that answer again)

672 (Double 2 is 4, 67−4=63, and 63÷7=9) Yes

105 (Double 5 is 10, 10−10=0, and 0 is divisible by 7) Yes

905 (Double 5 is 10, 90−10=80, and 80÷7=11 3/7) No

The last three digits are divisible by 8

109816 (816÷8=102) Yes

216302 (302÷8=37 3/4) No

A quick check is to halve three times and the result is still a whole number:

816/2 = 408, 408/2 = 204, 204/2 = 102 Yes

302/2 = 151, 151/2 = 75.5 No

The sum of the digits is divisible by 9

(Note: This rule can be repeated when needed)

1629 (1+6+2+9=18, and again, 1+8=9) Yes

2013 (2+0+1+3=6) No

The number ends in 0

220 Yes

221 No

Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.

1364 (+1−3+6−4 = 0) Yes

913 (+9−1+3 = 11) Yes

3729 (+3−7+2−9 = −11) Yes

987 (+9−8+7 = 8) No

The number is divisible by both 3 and 4 (it passes both the 3 rule and 4 rule above)

648
(By 3? 6+4+8=18 and 18÷3=6 Yes)
(By 4? 48÷4=12 Yes)
Both pass, so Yes

524
(By 3? 5+2+4=11, 11÷3= 3 2/3 No)
(Don't need to check by 4) No

There are lots more! Not only are there divisibility tests for larger numbers, but there are more tests for the numbers we have shown.

Factors Can Be Useful

Factors are the numbers you multiply to get another number:

This can be useful, because:

When a number is divisible by another number ...

... then it is also divisible by each of the factors of that number.

Example: If a number is divisible by 6, it is also divisible by 2 and 3

Example: If a number is divisible by 12, it is also divisible by 2, 3, 4 and 6

Another Rule For 11

Subtract the last digit from a number made by the other digits.
If that number is divisible by 11 then the original number is, too.

Can repeat this if needed,

28 − 6 is 22, which is divisible by 11, so 286 is divisible by 11

Example: 14641

1464 − 1 is 1463
146 − 3 is 143
14 − 3 is 11, which is divisible by 11, so 14641 is divisible by 11

1625, 1626, 1627, 1628, 2689, 3599, 3600, 3601, 3602, 5007

The divisibility rule of 9 states that if the sum of digits of any number is divisible by 9, then the number is also divisible by 9. It helps us in various concepts like finding divisors, HCF, LCM, measurements, and division. Divisibility by 9 is a rule that allows us to find whether a number is divisible by 9 or not without performing long division.

What is the Divisibility Rule of 9?

The divisibility rule of 9 helps us to find whether a number is a multiple of 9 or not without performing the actual division. Some of the multiples of 9 are 9, 18, 27, 36, 45, etc. Do you see a common pattern in the sum of the digits of these numbers? The sum of digits of all these numbers is itself a multiple of 9. For example, 18 is 1+8 = 9, which is divisible by 9, 27 is 2+7 = 9, which is divisible by 9, etc. So, as per the divisibility test of 9, if the sum of all the digits of a number is a multiple of 9, then the number is also divisible by 9.

Divisibility Rule of 9 Fun Activity

There is a fun activity based on the divisibility rule of 9. Ask your friend to think of any single-digit non-zero number. Ask her/him to take three times that number, make sure you should not know what is the number picked up by her/him. Now, tell her/him to multiply the result by 3. Now, ask her/him how many digits are there in the answer and tell you one of the digits from the resultant value. Then you can find out what the other digit of the number is, by using the divisibility test of 9. The other digit can be obtained by subtracting the known digit from 9. Let us try it out with a number 6. Three times 6 is 18. Now, multiply 18 by 3, which is 54. If we know any one of the digits, let's say 4, we can easily find out what is the other digit by subtracting it from 9, i.e., 9 - 4 = 5. So, the other digit is 5.

Divisibility Rule of 9 for Large Numbers

The divisibility rule of 9 remains the same for large numbers. The only difference is that we use the divisibility test of 9 repeatedly until we get the sum of the digits of the number closer to 9. For example, to find if 2374878 is divisible by 9 or not, we first find the sum of the digits, which is 2+3+7+4+8+7+8 = 39. Now, we will again add 3 and 9, which is 3+9 = 12, and 12 is not divisible by 9. So, 2374878 is not divisible by 9. Let us take another example. To find if 456318 is divisible by 9 or not, we first find the sum of the digits, which is 4+5+6+3+1+8 = 27. Now, we will again add 2 and 7, which is 2+7 = 9, and 9 is divisible by 9. So, 456318 is divisible by 9. Let us look at the steps to apply the divisibility rule of 9 easily with any large or smaller numbers:

Step 1: Find the sum of all the digits of the given number.
Step 2: Check if the sum is divisible by 9 or not. If it is still a large number, add the digits again.
Step 3: Check if the new sum is divisible by 9 or not. Repeat this process if you still find it difficult to figure out whether the sum of the digits is divisible by 9 or not.
Step 4: If the final sum is divisible by 9, then the original number would also be divisible by 9.

This is how the divisibility test of 9 works.

Divisibility by 3 and 9

Both the divisibility test of 9 and 3 are based on the same principle, which states that the sum of the digits of the given number should be divisible by them. To check if a number is divisible by 3 or not, the sum of all the digits of the number should be divisible by 3, while on the other hand in the case of divisibility rule by 9, if the sum of all the digits of the number is divisible by 9, then the number is also a multiple of 9.

For example, to find whether 459072 is divisible by 9 and 3 or not, let us find the sum of the digits. The sum is 4+5+9+0+7+2 = 27, which can be again summed to 2+7 = 9. The sum '9' is divisible by both 9 and 3, therefore, 459072 is divisible by both 9 and 3. Here, one important fact is that every number which is divisible by 9 is also divisible by 3 because 9 is itself a multiple of 3. On the other hand, every number which is divisible by 3 may or may not be divisible by 9.

Divisibility Rule of 9 and 11

The divisibility rule of 9 and 11 is different. As discussed earlier, the divisibility test of 9 says that the sum of the digits of the given number should be divisible by 9. However, the divisibility rule of 11 states that a number is divisible by 11 if the difference of the sum of the digits at even places and odd places is 0 or divisible by 11. So, we first find the difference of the sum of digits at even places and at odd places. It should be noted that both the rules are based on the sum of digits, but in the case of 11, we have to find the sum of digits at odd place values and at even place values separately, and then if the difference between the two sums is divisible by 11, the number will also be divisible by 11.

For example, let us find whether 99990 is divisible by 9 and 11 or not. The sum of all the digits is 9+9+9+9+0 = 36, which is divisible by 9, so 99990 is divisible by 9. Now, let us find the sum of the digits at even place values starting from the right, 9+9 = 18. The sum of digits at odd places from the right side is 0+9+9 = 18. Now, the difference between the two is 18-18 = 0, which is divisible by 11 (as 0 is divisible by every number). So, 99990 is divisible by both 9 and 11.

☛ Related Topics

Example 1: Using the divisibility rule of 9, state whether 724 is divisible by 9 or not.

Solution: Let us find the sum of all the digits of the number 724.

7+2+4 = 13

Here, 13 is not divisible by 9, so as per the divisibility test of 9, we can say that 724 is not divisible by 9.
Example 2: Check the divisibility of the following number by 9: 18972 without performing long division.

Solution: We can use the divisibility test of 9 here which states that if the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9. The sum of digits of 18972 is 1+8+9+7+2 = 27, which is divisible by 9. Therefore, we can say that 18972 is divisible by 9.
Example 3: Find the smallest 3-digit number divisible by 9.

Solution: The smallest 3-digit number is 100. In order to check the smallest number of 3 digits which can be a multiple of 9, we need to find the sum of the digits. Let us consider the units place as blank, so we have 10_. Now we have to find a digit that could come in the blank such that the sum of 1 and that digit is 9. That digit is 8, as 1+8 = 9. Therefore, 108 is the smallest 3-digit number which is divisible by 9.

go to slidego to slidego to slide

Breakdown tough concepts through simple visuals.

Math will no longer be a tough subject, especially when you understand the concepts through visualizations.

Book a Free Trial Class

FAQs on Divisibility Rule of 9

The divisibility rule of 9 states that if the sum of all the digits of a number is divisible by 9, then the number would be divisible by 9. It helps us to find whether 9 is a factor of any number or not without performing the actual division. For example, let us check if 85304 is divisible by 9. Since 8 + 5 + 3 + 0 + 4 = 20 and 20 is not divisible by 9, it can be said that 85304 is not divisible by 9.

What do the Divisibility Rules for 9 and 3 have in Common?

The divisibility test of 9 and 3 are based on the sum of the digits of the number. If the sum of the digits of the given number is divisible by 9 and 3, then the number will be divisible by 9 and 3 respectively. Note that all the numbers that are divisible by 9 are also divisible by 3, as 9 is itself a multiple of 3.

Using the Divisibility Rule of 9, Check if 1450 is Divisible by 9?

The sum of all the digits of 1450 is 1+4+5+0 = 10, which is not divisible by 9. So, 1450 is not divisible by 9, as per the divisibility test for 9.

How do you know if a Big Number is Divisible by 9?

With large numbers, we repeat the process of adding the digits of a number if we are not sure whether the sum of the digits is divisible by 9 or not. If that sum is divisible by 9, then the number is also divisible by 9. For example, to check whether 5409279 is divisible by 9 or not, we add all the digits, 5 + 4 + 0 + 9 + 2 + 7 + 9 = 36, which can be further added as 3 + 6 = 9, and 9 is divisible by 9. So, 5409279 is divisible by 9.

Using the Divisibility Rule of 9, Check if 8955 is Divisible by 9?

The sum of all the digits of 8955 is 8+9+5+5 = 27, which is divisible by 9. So, 8955 is divisible by 9, as per the divisibility rule for 9.

What is the Divisibility Rule of 3 and 9?

The divisibility rule of 3 states that if the sum of the digits of the given number is divisible by 3 then the number is divisible by 3. For example, let us check if 632 is divisible by 3. Since 6 + 3 + 2 = 11, and 11 is not divisible by 3, we can say that 632 is not divisible by 3. The divisibility rule of 9 states that if the sum of the digits of the given number is divisible by 9 then the number is divisible by 9. For example, let us check if 5499 is divisible by 9. Since 5 + 4 + 9 + 9 = 27, and 27 is divisible by 9, we can say that 5499 is divisible by 9.

Test Whether 9846 is Divisible by 9.

Using the divisibility rule of 9, let us check if 9846 is divisible by 9 or not. Since 9 + 8 + 4 + 6 = 27 and 27 is divisible by 9, we can say that 9846 is divisible by 9.