In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Maths > Playing With Numbers > Divisibility Tests
Playing With Numbers

Divisibility Tests

Suppose you’re at a stationary and you need to find which deal is better by using divisibility rules. Let’s say 2 pencils cost Rs 6 and in another store, 4 pencils cost Rs 8. Which deal is better? We know that 6 is divisible by 2, so each pencil costs Rs 3. Also, 8 is divisible by 4 which means each pencil cost Rs 2. We know that the shop with 4 pencils that cost Rs 8 is the best deal. Let us study the divisibility rules in detail.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Prime and Composite Numbers
Divisibility test for 2, 4 and 8
Divisibility test for 7
Slider

 

Divisibility Tests

You have bought 1235 chocolates and want to distribute them in your class. How will you know with what number to divide them without actually performing the operation of division? This is where the concept of divisibility rules comes into play.

The process of whether the given number is exactly divisible by another number without actually performing the operation of division is called the test of divisibility. Being exactly divisible means that on division there is no remainder left.  For understanding the divisibility rules for any number we need to know its divisibility with numbers like 2,3,4,5,9,10 and 11.

 Divisibility rules

(Source: youtube)

Browse more Topics Under Playing With Numbers

Divisibility Rules

Divisibility Test by 2

If the ones digit or unit’s place digit of a number is either 0, 2, 4, 6 0r 8  then the number is said to be divisible by 2. In other terms, if the last digit of the number is even then it is always divisible by 2. For example, let’s take  24. The last digit is even. 4 is divisible by 2, so the number is divisible by 2.

Divisibility Test by 4

A number is divisible by 4  if the number formed by the last two digits is divisible by 4. For example, let’s take 12343684. You don’t need to worry about all the numbers, just check the last 2 digits of the numbers. If the last two digits i.e 84 are divisible by 4than the number is divisible by 4.

Divisibility Test by 3 and 9

A number is divisible by 3 if the sum of its digits is divisible by 3. Let’s have a look at some examples.

  • 121: Here the sum of the digit is 4. Is 4=the number 4 divisible by 3? No.
  • 123:  Here the sum of the digit is 6 and yes 6 is divisible by 3. So the number 123 is divisible by 3.

Same rules are applied to the divisibility test of 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

Test of Divisibility by 5

A number is exactly divisible by 5 if it has the digits 0 or 5 at one’s place. The numbers like 15,120,205,4400 etc. are exactly divisible by 5 as these numbers have 0 or 5 in the one’s place. Let’s look at an example.

In the numbers 21345650 and 459022, which one is exactly divisible by 5? Out of both the numbers  21345650 and 459022, the first number i.e 21345650 is exactly divisible by 5 as the ones place is occupied by 0.

Test of Divisibility by 6

A number is exactly divisible by 6 if that number is divisible by 2 and 3 both. For checking the divisibility of the number with 6 we have to apply the rule of divisibility tests that we apply for numbers 2 and 3. Let’s look at an example. Using the divisibility tests rule for 6, check whether the following number is divisible by 6: 12,581.

In the number 12,581 the ones digit does not have any of the numbers like 0,2,4,6,8 in the ones place. So the number is not divisible by 2. Hence the number is not divisible by 6 because for a number to be divisible by 6, it has to be divisible by 2 and 3 both.

Test of Divisibility by 8

The test of divisibility by 8 is used in numbers with four or more digits. A number is exactly divisible by 8 if the number formed by the digits in ones, tens and hundredth place is divisible by 8. In the number 5864, we see that the number formed by the digits in ones, tens and hundredth place, 864 which is divisible by 8, so the number is divisible by 8.

For example, whether the following number is divisible by 8: 5,87,824. In the numbers given above, the number 5,87,824 has 824 in the ones, tens and hundredth place. This number is exactly divisible by 8, hence the number is divisible by 8.

Divisibility Test by 7

Test 1

Take the digits of the number in reverse order, from right to left, multiplying them successively by the digits 1, 3, 2, 6, 4, 5, repeating with this sequence of multipliers as long as necessary. Add the products. If the sum is divisible by 7 then it passes the divisibility test of 7.

Example:  Is the number 1603 divisible by 7?
Solution: Mutiplying by the respective digits, 3× 1 + 0 × 3 + 6 × 2 + 1 × 6= 21. This number is divisble by 7.

Test 2

Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary.

Example:  Is the number 1603 divisible by 7?
Solution: Applying the method, 160 – 2× 6 = 154. This number is divisible by 7.

Divisibility Test by 11

Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11 then so was the first number. Apply this rule over and over again as necessary. For example, let’s take the number 19151.

19151 → 1915 – 1 = 1914
1914 → 191 – 4 = 187
187 → 18 – 7 = 11
Hence, 19151 is divisible by 11.

Divisibility Rules for Higher Numbers

  • A number is exactly divisible by 10 if the one’s place of the number is occupied by 0. So, the numbers with 0 in the one’s place are divisible by 10.
  • A number is divisible by 13 if the number obtained by subtracting  9 times the last digit from the remaining digits of the number is divisible by 13.
  • A number is divisible by 17 if the number obtained after subtracting 5 times the last digit from the rest is divisible by 17.

Solved Examples for You

Question 1: Which of the following numbers is divisible by 14?

  1. 4683
  2. 7321
  3. 1428
  4. 5631

Answer: C is the correct option. The divisibility rule for 14 is that if the number is divisible by both 2 and 7 then the number is exactly divisible by 14. Here the last digit is, even so, the number 1428 is divisible by 2. 1428 is multiple of 7 so this number is also divisible by 7.

Question 2: Explain the divisibility rule of 2?

Answer:The divisibility rule for two tells us that any number whose last digit happens to be 0, 2, 4, 6, or 8 shall be divisible by 2. In other words, an even number is able to be divisible by 2. Also, an odd number will not be divisible by two.

Question 3: Explain the divisibility rule for 11?

Answer: The divisibility rule of eleven tells us that one must subtract and afterwards add the digits in an alternating pattern. This alternating pattern is from left to right. In case the answer is 0 or 11, then the result shall be divisible by 11.

Question 4: What is the importance of divisibility rules?

Answer: Divisibility rules are very useful as they help to quickly determine if a number can be divided by 2, 3, 4, 5, 9, and 10 without undertaking long division.

Question 5: Explain the divisibility rule of 5?

Answer: A number that is divisible by 5 should end in either 0 or 5. For this rule, let’s take a look at the last digit of 34,780. The last digit is a 0 which makes this number is divisible by 5.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

Leave a Reply

avatar
  Subscribe  
Notify of

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.