Divisibility Rules
Divisibility rules in math are a set of specific rules that apply to a number to check whether the given number is divisible by a particular number or not. Some known divisibility tests are for numbers 2 to 20. It helps us to find the factors and multiples of numbers without performing long division. A person can mentally check whether a number is divisible by another number or not by applying divisibility rules. Let us learn more about divisibility tests in this article.
1.  What Are Divisibility Rules? 
2.  Divisibility Rules From 2 to 12 
3.  Divisibility Rules for Prime Numbers 
4.  FAQs on Divisibility Rules 
What are Divisibility Rules?
A divisibility rule is a kind of shortcut that helps us to identify if a given integer is divisible by a divisor by examining its digits, without performing the whole division process. Multiple divisibility rules can be applied to the same number which can quickly determine its prime factorization. A divisor of a number is an integer that completely divides the number without leaving any remainder.
In a 1962 Scientific American article, the popular mathematics and science writer, Martin Gardner, discussed divisibility rules for 2–12, where he explains that the rules were widely known during the renaissance and used to reduce fractions with large numbers down to the lowest terms. Since every number is not completely divisible by every other number, they may leave a remainder other than zero. There are certain rules which help us determine the actual divisor of a number just by considering the digits of that number. These are called divisibility rules.
Divisibility Rules From 2 to 12
In this section, we will learn about basic divisibility tests from 2 to 12. The divisibility rule of 1 is not required since every number is divisible by 1. Here are a few basic divisibility rules:
Divisibility by number  Divisibility Rule 

Divisible by 2  A number that is even or a number whose last digit is an even number i.e. 0, 2, 4, 6, and 8. 
Divisible by 3  The sum of all the digits of the number should be divisible by 3. 
Divisible by 4  Number formed by the last two digits of the number should be divisible by 4 or should be 00. 
Divisible by 5  Numbers having 0 or 5 as their ones place digit. 
Divisible by 6  A number that is divisible by both 2 and 3. 
Divisible by 7  Subtracting twice the last digit of the number from the remaining digits gives a multiple of 7. 
Divisible by 8  Number formed by the last three digits of the number should be divisible by 8 or should be 000. 
Divisible by 9  The sum of all the digits of the number should be divisible by 9. 
Divisible by 10  Any number whose one's place digit is 0. 
Divisible by 11  The difference of the sums of the alternative digits of a number is divisible by 11. 
Divisible by 12  A number that is divisible by both 3 and 4. 
Divisibility Rules Chart and Examples
Let's try to understand the above divisibility tests with examples.
 Is 280 divisible by 2? Yes, 280 is divisible by 2 as the unit's place digit is 0.
 Is 345 divisible by 3? Yes, 345 is divisible by 3, as the sum of all the digits i.e. 3 + 4 + 5 = 12, and 12 is divisible by 3. So, 345 is divisible by 3.
 Is 450 divisible by 4? No, 450 is not divisible by 4 as the number formed by the last two digits starting from the right, i.e 50 is not divisible by 4.
 Is 3900 divisible by 5? Yes, 3900 is divisible by 5 as the digit at the unit's place is 0 which satisfies the divisibility rule of 5.
 Is 350 divisible by 6? The sum of all the digits of 350 is 8, so it is not divisible by 3. Hence it cannot be divisible by 6, as a number needs to be a common multiple of both 2 and 3 to be a multiple of 6.
 357 is divisible by 7 as when we subtract the twice of the ones place digit, 7 × 2 = 14, and subtract it from the remaining digits 35, we get 35 14 = 21, which is divisible by 7. So, 357 is divisible by 7.
 79238 is not divisible by 8, as the number formed by the last three digits 238 is not completely divisible by 8.
 875 is not divisible by 9, as the sum of all the digits, 8 + 7 + 5 = 20 is not divisible by 9.
Now, let us take the number 1000 and see its divisibility by 2 to 10. It is clearly seen in the image that 1000 is divisible by 2, 4, 5, 8, and 10, and not divisible by 3, 6, 7, and 9. We find this by applying the divisibility rules of 2 to 10, and not by performing division which can be more timeconsuming.
Divisibility Rules for Prime Numbers
Intermediate divisibility rules are applied to prime numbers which are less than 20 and greater than 10. Divisibility tests for prime numbers 2, 3, 5, 7, and 11 are already discussed above. Here, let's learn about the divisibility rules of 13, 17, and 19.
Divisibility Rule of 13  A number is divisible by 13 when it leaves 0 as the remainder when we divide it by 13. The divisibility test of 13 helps us to quickly find out whether a number is divisible by 13 or not without performing long division. According to the divisibility rule of 13, first, we have to multiply the ones place digit by 4. Then, we add the product to the rest of the number to its left (excluding the digit at the unit's place). If that sum results in a number divisible by 13, then the original number is also divisible by 13. Apart from this method, there are three other divisibility rules of 13 that are explained in this article  Divisibility Rule of 13. Have a look!
Divisibility Rule of 17  A number is divisible by 17 when 17 divides it completely without leaving any nonzero remainder. According to the divisibility rule of 17, first, we have to multiply the ones place digit by 5. Then, we subtract the product from the rest of the number to its left (excluding the digit at the unit's place). If that difference results in a number divisible by 17, then the original number is also divisible by 17.
Divisibility Rule of 19  If we get 0 as the remainder when dividing a number by 19, then that number is considered divisible by 19. According to the divisibility rule of 19, first, we have to multiply the ones place digit by 2. Then, we add the product to the rest of the number to its left (excluding the digit at the unit's place). If that sum results in a number divisible by 19, then the original number is also divisible by 19.
Divisibility Rules of 13, 17, and 19 Examples
Let us take the example of number 1326, and check its divisibility by 13, 17, and 19. Have a look at the image given below.
Challenging Questions on Divisibility Tests
 A number is divisible by 4 and 12. Is it true that it will be divisible by 48?
 Check whether 2359334 follows divisibility rules of 4 as well as 8.
Divisibility Rules Tips and Tricks:
 Divisibility rules are of great importance while checking prime numbers.
 These are handy to solve word problems.
 They are useful to do quick calculations.
 Every even number is divisible by 2.
 Every leap year is divisible by 4.
Divisibility Tests of Numbers
Also, check these articles related to the divisibility rules.
Divisibility Rules with Examples

Example 1: Sam collected 156 cans for an Art project. Kim also collected a few more cans. Together the total number of cans represents a number that is not divisible by 2. Which of the following represents the number of cans they collected together?
Select an option: 236, 254, 289, 278
Solution:
As per the divisibility rule of 2, a number is divisible by 2 if it is even or if the last digit is an even number, i.e. 2, 4, 6, 8 including 0. Thus, out of the listed numbers 236, 254, 289, and 278, only 289 is not divisible by 2. This indicates that they had collected 289 cans together.

Example 2: Mr. Markson sells tickets for film festivals. The table below shows the number of tickets he sold for 2 days last month. Which of the following statements is true?
Days Number of tickets sold Monday 396 Tuesday 169 (a) The number of tickets sold on Tuesday is divisible by 13
(b) The number of tickets sold on Monday is divisible by 3, 6, and 9
Solution:
(a) Using the divisibility rules of 13, we can say that 16 + (9 × 4) = 52 is divisible by 13. Therefore, the number of tickets sold on Tuesday is divisible by 13. The given statement is true.
(b) The number of tickets sold on Monday is 396. The sum of digits of 396 = 3 + 9 + 6 = 18. ∵ 18 is divisible by 3 ⇒ 396 is also divisible by 3.
We know 396 is an even number. Hence, it is divisible by 2. ∵ It is divisible by both 2 and 3, ⇒ 396 is also divisible by 6.As the sum of the digits is 18 which is divisible by 9, this implies 396 is divisible by 9. Therefore, 396 is divisible by 3, 6, and 9. The given statement is true.
Therefore, both (a) and (b) statements are true.

Example 3: Flynn is arranging some photographs for his family book. He needs to put the same number of photographs in each row with no photographs left over. If he has 288 photographs to arrange, how many photographs can he place in each row? Is it 5, 9, or 10?
Solution:
To find out the number of photos to be placed in each row, we need to check whether 288 is divisible by 5, 9, or 10 by using divisibility rules. The ones place digit of 288 is 8, so it is not divisible by 5 and 10, as a number should have 0 or 5 at its ones place digit to be divisible by 5, and it should have 0 at the units place to be divisible by 10. Now, let us check its divisibility by 9. The sum of the digits of the given number is 2+8+8=18, which is divisible by 9, so 288 is divisible by 9.
Checking its divisibility by 5, 9, and 10, we can say that 288 is divisible by 9. Hence, 9 photographs can be placed in each row.
FAQs on Divisibility Rules
What is the Meaning of Divisibility Rules?
Divisibility rules help us to identify whether or not a number is completely divisible by another number. If a number 'a' is divisible by another number 'b', then it is denoted as 'ab'. Divisibility tests are very short calculations based on the digits of the numbers to find out if a particular number is dividing another number completely or not.
What is the Divisibility Rule of 7 and 11?
The divisibility rule of 7 states that if we multiply the units place digit of the number by 2, and then if the difference between that number and the rest of the number to the left is divisible by 7, then the number is also divisible by 7. For example, let us check whether the number 3437 is divisible by 7 or not. First, find the twice of the ones place digit, i.e 7. Now, subtract 7×2=14 from the rest of the number on the left, which is 343. 343  14 = 329. It is still difficult to figure out whether 329 is divisible by 7 or not, so repeat the same process again. Subtract 9×2=18 from 32, we get 3218=14, which is divisible by 7. So, 3437 is divisible by 7.
The divisibility rule of 11 states that if the difference between the sums of the digits at the alternative places of a number is divisible by 11, then the number is also divisible by 11. To check if 1334 is divisible by 11 or not, find the sum of the digits at the alternative places first. The sum of the digits at the odd places is 4+3=7 and the sum of the digits at the even places is 3+1=4. Now find the difference between the two, which is 74=3. 3 is not divisible by 11, so 1334 is also not divisible by 11.
What are the Divisibility Rules for 2, 5, and 10?
The divisibility rules of 2, 5, and 10 are given below:
 Divisibility rule of 2  Units place digit of the number should be either 0, 2, 4, 6, or 8.
 Divisibility rule of 5  Units place digit of the number should be either 0 or 5.
 Divisibility rule of 10  Units place digit of the number should be 0.
What are the Divisibility Rules for 3, 6, and 9?
The divisibility rules of 3, 6, and 9 are given below:
 Divisibility rule of 3  The sum of all the digits of the number should be divisible by 3.
 Divisibility rule of 6  Number should be divisible by both 2 and 3.
 Divisibility rule of 9  The sum of all the digits of the number should be divisible by 9.
What are the Divisibility Rules for 8?
To check whether a number is divisible by 8 or not, we can use the divisibility test of 8 which states that for a number to be divisible by 8 either of the following should be true:
 The last three places of the number from the right should be 000.
 The last three places of the number should be a number divisible by 8.
What is the Divisibility Test of 7?
Look at the steps given below to apply the divisibility test of 7:
 Step 1: Identify the ones place digit of the number and multiply it by 2.
 Step 2: Find the difference between the number obtained in step 1 and the rest of the number.
 Step 3: If the difference is divisible by 7, then the number is divisible by 7.
 Step 4: If it is still difficult to identify whether the difference is a multiple of 7 or not, repeat the same process with the number obtained in step 2.
What is the Divisibility Test of 2?
The divisibility test of 2 states that if the ones place digit of a number is even including 0, then the number will be divisible by 2. All the even numbers are divisible by 2, or we can say are multiples of 2.
What is the Use of Divisibility Rules?
In math, divisibility tests are important to learn as it helps us to ease out our calculations where we have to do multiplication and division. We can quickly identify whether a particular number is divisible by another number or not by applying divisibility rules.
How many Divisibility Rules are there?
Generally, we have divisibility rules from 1 to 20. But if we would able to identify the pattern of multiples of numbers, we can create more divisibility tests. For example, the divisibility rule of 21 states that a number must be divisible by both 3 and 7. It is because 21 is a multiple of two prime numbers 3 and 7, so all the multiples of 21 will definitely have 3 and 7 as their common factors.
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