Divisibility Rules

What are Divisibility Rules?

Divisibility rules are simple mathematical shortcuts that help determine whether a number can be divided exactly by another number without performing the actual division. These rules are widely used in arithmetic, number systems, factorization, HCF and LCM, and quantitative aptitude. Learning divisibility rules improves mental calculation speed and helps solve mathematical problems more efficiently.

Divisibility Rules Meaning in Simple Words

Divisibility rules are quick tests that tell us whether a number is divisible by another number without carrying out long division.

For example:

• 246 is divisible by 2 because its last digit is 6.
• 351 is divisible by 3 because the sum of its digits is 9.
• 875 is divisible by 5 because it ends in 5.

These simple checks save time and simplify calculations.

Definition of Divisibility Rules

A divisibility rule is a mathematical condition used to determine whether a number can be divided by another number exactly, leaving no remainder.

If a number satisfies the required condition, then it is divisible by that divisor.

Example

Consider the number:

$468$

Since:

$4+6+8=18$

and 18 is divisible by 3,

therefore 468 is divisible by 3.

Similarly,

the last digit of 468 is 8, which is even.

Therefore, 468 is also divisible by 2.

Why Divisibility Rules are Important

Divisibility rules are important because they simplify calculations and help solve number system questions quickly.

Importance of Divisibility Rules

• Improve mental mathematics skills.
• Reduce the need for lengthy division calculations.
• Help in finding factors and multiples.
• Useful for HCF and LCM calculations.
• Assist in prime factorization.
• Save time in competitive examinations.
• Improve accuracy in arithmetic operations.

Competitive Exam Relevance

Questions based on divisibility rules frequently appear in:

• SSC CGL
• SSC CHSL
• Banking Exams
• CUET
• CAT
• Railways
• NDA
• CDS
• State PSC Exams

Real-Life Applications of Divisibility Rules

Divisibility rules are not limited to classroom mathematics; they are also useful in everyday calculations.

Examples

• 240 chocolates can be equally distributed among 8 students because 240 is divisible by 8.
• A pack of 150 notebooks can be arranged into groups of 5 because it is divisible by 5.
• A warehouse manager can quickly determine whether 432 items can be packed into boxes of 9 using divisibility rules.

Divisibility Rules Chart

A divisibility rules chart provides a quick reference for checking whether a number is divisible by common divisors. Memorizing these rules can significantly improve speed in arithmetic and aptitude questions.

Divisibility Rules from 1 to 20

The following table summarizes the divisibility rules for numbers from 1 to 20.

Quick Divisibility Rules Table

The most frequently used divisibility tests are summarized below.

These rules are among the most commonly tested in quantitative aptitude and number system questions.

Rules of Divisibility or Divisibility Test in Maths

In divisibility rules(test) we find whether a given number is divisible by another number, we perform actual division and see whether the remainder is zero or not.

In mathematics, there are different rules of divisibility for every number. In this article, we will learn about them in detail.

Divisibility rule of 1

Every number is divisible by 1. So there is no particular divisibility rule for 1.

Example: 2 is divisible by 1, 100 is divisible by 1, etc.

Divisibility rule of 2

If the last digit of a number is even (i.e., 0, 2, 4, 6, 8), then the number is divisible by 2.

Example: 12 is divisible by 2, 24 is divisible by 2, 66 is divisible by 2, etc.

Divisibility rule of 3

If the sum of its digits of a number is divisible by 3, then that number is divisible by 3.

Example: 123 is divisible by 3 as the sum of the digits 1 + 2 + 3 = 6 is divisible by 3.

768 is divisible by 3 as the sum of the digits 7 + 6 + 8 = 21 is divisible by 3.

Divisibility rule of 4

If the last two digits of a number from left to right are divisible by 4, then that number is divisible by 4.

Example: 2064 is divisible by 4 as 64 is divisible by 4.

1532 is divisible by 4 as 32 is divisible by 4.

There is another way to check if a number is divisible by 4 or not. If the last two digits of a number are 0, then that number is divisible by 4.

Example: 600 is divisible by 4, 700 is divisible by 4, 1300 is divisible by 4, etc.

Divisibility rule of 5

If the last digit of a number is 0 or 5, then that number is divisible by 5.

Example: 125 is divisible by 5, 50 is divisible by 5, 575 is divisible by 5, etc.

Divisibility rule of 7

For 7 divisibility rule states that if a number is divisible by 7, then “the difference between twice the unit digit of the given number and the remaining part of the given number should be a multiple of 7 or it should be equal to 0”.

We have to take an example to understand it better.

Let's check if 2839155 is divisible by 7.

The unit digit of the number is 5.

So, 283915 – 5 × 2 = 283905

28390 – 5 × 2 = 28380

2838 – 0 = 2838

283 – 8 × 2 = 267

26 – 7 × 2 = 12, which is not divisible by 7.

So, the number is not divisible by 7.

Divisibility rule of 8

If the last three digits of a number from left to right are divisible by 8, then that number is divisible by 8.

Example: 151629664 is divisible by 8 as 664 is divisible by 8.

Divisibility rule of 9

For 9 divisibility rule states that if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Example: 7363927 is divisible by 9 as the sum of the digits 7 + 3 + 6 + 3 + 9 + 2 + 6 = 36 is divisible by 9.

Divisibility rule of 10

If the last digit of a number is 0, then that number is divisible by 10.

Example: 100, 520, 630, 1000, etc.

Divisibility rule of 11

A number is divisible by 11 if the difference between the sum of the digits at even places and the sum of the digits at odd places of a number is either 0 or divisible by 11.

Example: 1726628 is not divisible by 11 as (1 + 2 + 6 + 8) - (7 + 6 + 2) = 2 is not divisible by 11.

3460831 is divisible by 11 as (3 + 6 + 8 + 1) - (4 + 0 + 3) = 11 is divisible by 11.

Divisibility rule of 13

The 13 divisibility rule states that to check if a number is divisible by 13, first, multiply the last digit/unit digit of the number by 4 and subtract it from the remaining number. Then check if it is divisible by 13. For larger numbers, use this process until you get a smaller number to check if it is divisible by 13 or not.

Let's check if 2639155 is divisible by 13.

The unit digit of the number is 5.

So, 263915 – 5 × 4 = 263895

26389 – 5 × 4 = 26369

2636 – 9 × 4 = 2600

260 – 0 × 4 = 260

26 – 0 × 4 = 26, which is divisible by 13.

So, the number is divisible by 13.

Divisibility rule of 16

If the last 4 digits of a number from left to right are divisible by 16, then the whole number is divisible by 16.

Example: 162652432 is divisible by 16 as 2432 is divisible by 16.

Divisibility rule of 17

First, multiply the last digit/unit digit of the number by 5 and subtract it from the remaining number. Check if it is divisible by 17. For larger numbers, use this process until you get a smaller number to check if it is divisible by 17 or not.

Let's check if 109718 is divisible by 17.

The unit digit of the number is 8.

So, 10971 – 5 × 8 = 10931

1093 – 5 × 1 = 1088

108 – 5 × 8 = 68, which is divisible by 17.

So, the number is divisible by 17.

Divisibility rule of 19

First, double the last digit of the number and add it to the remaining number. Check if it is divisible by 19. For larger numbers, use this process until you get a smaller number to check if it is divisible by 19 or not.

Let's check if 2639155 is divisible by 7.

The unit digit of the number is 5.

So, 263915 + 5 × 2 = 263925

26392 + 5 × 2 = 26402

2640 + 2 × 2 = 2644

264 + 6 × 2 = 266

26 + 6 × 2 = 38, which is divisible by 19.

So, the number is divisible by 19.

Commonly Used Divisibility Rules

Some divisibility rules are used more frequently than others because they appear regularly in arithmetic calculations and competitive examinations.

Divisibility Rule of 2

A number is divisible by 2 if its last digit is:

$0,\ 2,\ 4,\ 6,\ 8$

Examples:

$248,\ 356,\ 1240$

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example:

$531$

$5+3+1=9$

Since 9 is divisible by 3,

531 is divisible by 3.

Divisibility Rule of 5

A number is divisible by 5 if it ends in:

$0 \text{ or } 5$

Examples:

$250,\ 485,\ 1200$

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Example:

$729$

$7+2+9=18$

Since 18 is divisible by 9,

729 is divisible by 9.

Divisibility Rule of 10

A number is divisible by 10 if its last digit is 0.

Examples:

$120,\ 540,\ 2500$

How to Memorize Divisibility Rules

Memorizing divisibility rules becomes easier when similar rules are grouped together and practiced regularly.

Easy Memorization Techniques

• Remember that rules for 3 and 9 are based on the sum of digits.
• Remember that rules for 2, 5, and 10 depend on the last digit.
• Remember that rules for 4 and 8 depend on the last two or three digits.
• Remember that 6 requires both divisibility by 2 and 3.
• Remember that 12 requires both divisibility by 3 and 4.
• Practice divisibility checks on everyday numbers.
• Use flashcards or quick revision tables.

Quick Memory Pattern

By learning these patterns instead of memorizing every rule separately, students can quickly master divisibility rules and solve number system questions with greater speed and accuracy.

Understanding the divisibility rule of 13, 17, and 19 with examples

The divisibility rule of 13, 17, and 19 is a bit complicated to remember. So we will take an example and check the same number is divisible by 13, 17, and 19.

Best Books for Divisibility Rules

A strong understanding of divisibility rules helps improve mental calculations, factorization, number systems, and quantitative aptitude problem-solving. These books provide concept clarity and extensive practice.

Shortcut Tips and Tricks for Divisibility Rules

Learning divisibility shortcuts can significantly improve calculation speed and help solve number system questions without lengthy division. These tricks are especially useful in aptitude and competitive examinations.

Important Divisibility Rules Table

This table provides a quick reference to the most commonly used divisibility tests. Memorizing these rules can save time and improve accuracy while solving arithmetic and number system problems.

Practice Questions based on Divisibility Rules

Q1. What is the smallest number that can be added to 9454351626 so that it becomes divisible by 11?

1. 1

2. 6

3. 5

4. 4

Hint: A number is divisible by 11 if the difference between the sum of the digits at even places and the sum of the digits at odd places of a number is either 0 or divisible by 11.

Answer:

A number is divisible by 11 if the difference between the sum of the digits at even places and the sum of the digits at odd places of a number is either 0 or divisible by 11.

Now, the difference between the sums = (9 + 5 + 3 + 1 + 2 − 4 − 4 − 5 − 6 − 6) = − 5

⇒ So, the difference is 5.

So, either we have to subtract 5 or add (11 – 5) = 6 to make 9454351626 divisible by 11.

Therefore, the smallest number that can be added to 9454351626 to make it divisible by 11 is 6.

Hence, the correct answer is 6.

Q2. Find the value of 'a' to make 6234a6 divisible by 9.

1. 8

2. 7

3. 10

4. 6

Hint: A number is divisible by 9 only when the sum of the digits of the number is divisible by 9.

Answer:

Given: The number is 6234a6.

A number is divisible by 9 when the sum of its digits is divisible by 9.

The sum of the digits of 6234a6 = 6 + 2 + 3 + 4 + a + 6 = 21 + a

The nearest value greater than 21 and divisible by 9 is 27.

So, 21 + a = 27

⇒ a = 6

Hence, the correct answer is 6.

Q3. Which of the following numbers is divisible by 8?

1. 18718

2. 18716

3. 18712

4. 18714

Hint: If the last three digits of a number are divisible by 8, then the number is divisible by 8.

Answer:

If the last three digits of a number are divisible by 8, then the number is divisible by 8.

Option 1: 18718. 718 cannot be divided by 8.

Option 2: 18716. 716 cannot be divided by 8.

Option 3: 18712. 712 is divisible by 8. Therefore, 18712 is also divisible by 8.

Option 4: 18714. 714 cannot be divided by 8.

Hence, the correct answer is 18712.

Q4. $4^{11}+4^{12}+4^{13}+4^{14}$ is divisible by:

1. 7

2. 14

3. 17

4. 9

Hint: Take $4^{11}$ common from the given expression.

Answer:

Given:

$4^{11}+4^{12}+4^{13}+4^{14}$

= $4^{11}(1+4+4^2+4^3)$

= $4^{11}(85)$

Since 85 is divisible by 17,

Therefore, the above expression is also divisible by 17.

Hence, the correct answer is 17.

Q5. Which of the following numbers is divisible by 99?

1. 31548

2. 60687

3. 44775

4. 84456

Hint: You have to check if the number is divisible by 9 and 11 to solve it.

Answer:

If a number is divisible by 99, then it must be divisible by 11 and 9, because 99 = 11 × 9

To check the divisibility by 11 we have to check the difference of the sum of odd place digits and the sum of even place digits.

If the result is 0 or a multiple of 11, then the number is divisible by 11.

To check the divisibility by 9, the sum of digits must be multiple of 9.

60687: Sum of odd place digits – Sum of even place digits = (6 + 6 + 7) – (0 + 8) = 11, which is divisible by 11.

Also, the sum of all digits = 6 + 0 + 6 + 8 + 7 = 27, which is divisible by 9.

So, 60687 will be divisible by 99.

Hence, the correct answer is 60687.

Q6. XY7B is a 4-digit number divisible by 4. What is the largest value of B?

1. 6

2. 2

3. 0

4. 8

Hint: For a number to be divisible by 4, the last 2 digits should be divisible by 4.

Answer:

For a number to be divisible by 4, the last 2 digits should be divisible by 4.

So, 7B should be divisible by 4.

76 is the largest number of the form 7B, which is divisible by 4.

Hence, the correct answer is 6.

Q7. The number 2918245 is divisible by which of the following numbers?

1. 3

2. 11

3. 12

4. 9

Hint: Divisibility rule for 3: A number is completely divisible by 3 if the sum of its digits is divisible by 3.

Divisibility rule for 11: The difference between the sum of digits at the odd position and the sum of digits at the even position in a number is 0 or 11.

Answer:

Given number: 2918245

Divisibility rule for 3: A number is completely divisible by 3 if the sum of its digits is divisible by 3.

Here, 2 + 9 + 1 + 8 + 2 + 4 + 5 = 31, which is not divisible by 3

⇒ The given number will not be divisible by 12 and 9 either.

Divisibility rule for 11: The difference between the sum of digits at the odd position and the sum of digits at the even position in a number is 0 or 11.

⇒ (2 + 1 + 2 + 5) – (9 + 8 + 4)

= 10 – 21

= –11, which is divisible by 11.

Hence, the correct answer is 11.

Q8. If the four-digit number 463y is divisible by 7, then what is the value of y?

1. 4

2. 6

3. 3

4. 5

Hint: First, divide the number by 7 and find the remainder. Then equate the remainder with 7 to find y.

Answer:

463y is divisible by 7.

463y = (7 × 661) + 3 + y

7 = 3 + y

⇒ y = 7 – 3 = 4

Hence, the correct answer is 4.

Q9. If the number 647592 is divisible by 88 and if the digits are rearranged in increasing order, then the new number thus formed will be divisible by:

1. 22

2. 66

3. 44

4. 3

Hint: If the sum of all digits of a number is divisible by 3 then the number will be divisible by 3.

Answer:

Given: The number 647592 is divisible by 88.

Rearranging the number in increasing order ⇒ 245679

Now, the sum of the digits = 2 + 4 + 5 + 6 + 7 + 9 = 33, which is divisible by 3.

So, the new number will be divisible by 3.

Hence, the correct answer is 3.

Q10. How many of the following numbers are divisible by 156?

312, 620, 936, 1402, 1872, 3216, 7176, 8108.

1. 5

2. 3

3. 4

4. 2

Hint: When we divide a number if we get the remainder as 0 then we can say that the given number is divisible by the divisor.

Answer:

Given numbers: 312, 620, 936, 1402, 1872, 3216, 7176, 8108.

312 = 156 × 2 + 0

620 = 156 × 3 + 152

936 = 156 × 6 + 0

1402 = 156 × 8 + 154

1872 = 156 × 12 + 0

3216 = 156 × 20 + 96

7176 = 156 × 46 + 0

8108 = 156 × 51 + 152

$\therefore$ A total of 4 numbers are divisible by 156.

Hence, the correct answer is 4.

Q11. If the 9-digit number $72 x 8431y 4$ is divisible by 36, what is the value of $(\frac{x}{y}-\frac{y}{x})$ for the smallest possible value of $y$, given that $x$ and $y$ are natural numbers?

1. $1 \frac{5}{7}$

2. $2 \frac{1}{10}$

3. $1 \frac{2}{5}$

4. $2 \frac{9}{10}$

Hint: First, check the divisibility of 4 and 9 and solve for $x$ and $y$. Then solve the required expression using it.

Answer:

If $72 x 8431y 4$ is divisible by 36,

Then, $72 x 8431y 4$ is divisible by 4 and 9.

Since it is divisible by 4, $y4$ is divisible by 4.

On putting $y=2, y4 = 24$ which is divisible by 4.

$\therefore$ The smallest possible value of $y$ is 2.

Now $72 x 8431y 4$ becomes $72 x 84312 4$

As it is divisible by 9,

$7+2+x+8+4+3+1+2+4=31+x$ is divisible by 9

⇒ $x=5$

$\therefore x=5, y=2$

$(\frac{x}{y}-\frac{y}{x})=(\frac{5}{2}-\frac{2}{5})=\frac{25-4}{10}=\frac{21}{10}=2\frac{1}{10}$

Hence, the correct answer is $2\frac{1}{10}$.

Q12. The number 150328 is divisible by 23. If the digits are rearranged in descending order and five times 13 is subtracted from the new number thus formed, then the resultant number will be divisible by:

1. 3

2. 5

3. 11

4. 2

Hint: The number 150328 is divisible by 23. When rearranged in descending order we get 853210. Now subtract 65 and then find out the number it can be divided by.

Answer:

Here, the number 150328 is divisible by 23.

When rearranged in descending order we get = 853210

When 5 times 13 i.e., 65 is subtracted from the new number, we get = 853210 – 65 = 853145

$\therefore$ The resultant number is divisible by 5 since the last digit is 5.

Hence, the correct answer is 5.

Q13. What are the values of R and M, respectively, if the given number is perfectly divisible by 16 and 11?

34R05030M6

1. 4 and 6

2. 7 and 5

3. 5 and 5

4. 5 and 7

Hint: A number is divisible by 16 if and only if the last 4 digits of the number are divisible by 16. A number is divisible by 11 if the difference of the sum of odd and even places is divisible by 11.

Answer:

Any number that has 4 or greater than 4 digits is divisible by 16 if and only if the last 4 digits of the number are divisible by 16.

Last 4 digits = 30M6

$\frac{30M6}{16}$ should be a whole number.

So M = 5 because $\frac{3056}{16}$ = 191

Now the number is 34R0503056

A number is divisible by 11 if the difference of the sum of odd and even places is divisible by 11.

To be divisible by 11,

$\frac{[(3 + R + 5 + 3 + 5) - (4 + 0 + 0 + 0 + 6)]}{11}$ should be a whole number

$=\frac{16+R-10}{11}$

$=\frac{6+R}{11}$

Hence, R must be 5 so that (6 + R) is divisible by 11.

Hence, the correct answer is 5 and 5.

Q14. If a 4-digit number x58y is exactly divisible by 9, then the least value of (x + y) is:

1. 4

2. 5

3. 3

4. 2

Hint: Divisibility rule of 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

Answer:

A number is divisible by 9 if the sum of its digits is divisible by 9.

Given a 4-digit number x58y, the sum of its digits is x + 5 + 8 + y = (x + y) + 13.

For the number to be divisible by 9, (x + y) + 13 must be divisible by 9.

The smallest possible value for (x + y) that makes (x + y) + 13 divisible by 9 is 5 (since 5 + 13 = 18, and 18 is the multiple of 9).

Hence, the correct answer is 5.

Q15. If the number 476xy0 is divisible by both 3 and 11, then in the hundredth and tenth places, the non-zero digits are, respectively:

1. 2 and 3

2. 3 and 2

3. 5 and 8

4. 8 and 5

Hint: A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or divisible by 11.

Answer:

Let the given number be 476xy0.

Then (4 + 7 + 6 + x + y + 0) = (17 + x + y) must be divisible by 3.

And, (0 + x + 7) − (y + 6 + 4) = (x − y − 3) must be either 0 or 11.

From these conditions, we get x = 8 and y = 5.

Hence, the correct answer is 8 and 5.

Related Quantitative Aptitude Topics

Strengthen your quantitative aptitude preparation by exploring related mathematics topics that build numerical ability, logical thinking, and problem-solving skills. These concepts are frequently asked in competitive exams and help develop a strong foundation in arithmetic, algebra, and number systems.