Aptitude Made Simple
Numbers – Divisibility Test
What is Divisibility Test?
When you look at any number whether it is 2 digit or 4 digit or 10 digit number, without actually dividing that number with 2 ,3, 4(any number) if you can tell that number is divisible by 4 completely that is nothing but divisibility Test.
Let us take sample example to understand it and we will go through divisibility tests for 2 to 11.
Sample example
Write your mobile on piece of paper. I have written mine below:
9096754428 Now looking at number tell if this number is completely divisible by 9 or not?
You will now try to divide complete number by 9 and check what remainder is. If remainder is 0 means completely divisible. Correct?
Now just do simple thing: Add all digits of given number
Here: 9 + 0 + 9 + 6 + 7 + 5 + 4 + 4 + 2 + 8 = 54
Now check if 54 is completely divisible by 9 or not.
Yes 54 is divisible by 9
So, The number 9096754428 is also completely divisible by 9
(This is nothing but divisibility test of 9 and each number has different divisible test) Why Divisibility Test?
It simplifies your calculation. It also changes the way we look at numbers. When you do regular practice of divisibility test (2 to 11 numbers) and you look at any number you will think whether it is divisible by 2, 3 , 4 …11 etc and will make your calculations quick and short.
You will see lot of benefits of this in all different topics while solving in it.
Benefits of Divisibility Tests:
HCF and LCM | Simplification |
Work and Time | Average |
Pipes and Cistern | Series |
Quick calculation and any topic where quick calculations are required | |
Divisibility Test for 2:
Rightmost digit number should be divisible by 2.
All even numbers are completely divisible by 2.
In simple words, if rightmost last digit of any number is 0 or 2 or 4 or 6 or 8 then that number is completely divisible by 2
Look at below numbers and observe:
Number | Observation [right most digit] | Divisible by 2 or not |
35 | Rightmost Last digit is 5 and 5 is not divisible by 2 | Number is not divisible by 2 |
70 | Rightmost Last digit is 0 and 0 is divisible by 2 | Number is divisible by 2 |
269 | Rightmost Last digit is 9 and 9 is not divisible by 2 | Number is not divisible by 2 |
1002 | Rightmost Last digit is 2 and 2 is divisible by 2 | Number is divisible by 2 |
Problem 1:
Number 3789* is completely divisible by 2. What number could be there in place of * ? Solution :
As per divisibily test for 2, any number whose rightmost digit is 0 or 2 or 4 or 6 or 8 is completely divisible by 2.
So in place of * there could be below values:
2, 4, 6, 8, 0
Answer is 2, 4, 6, 8, 0
Problem 2 :
Which of the following numbers are completely divisible by 2 ?
1236665, 888886, 9999995, 9876543
Solution :
As per divisibility test for 2, any number whose rightmost digit is divisible by 2 is completely divisible by 2.
So let us look at right most number and decide if number is divisible by 2 or not.
Number | Observation | Divisible by 2 or not |
1236665 | Rightmost Last digit is 5 and 5 is not divisible by 2 | Number is not divisible by 2 |
888886 | Rightmost Last digit is 6 and 6 is divisible by 2 | Number is divisible by 2 |
9999995 | Rightmost Last digit is 5 and 5 is not divisible by 2 | Number is not divisible by 2 |
9876543 | Rightmost Last digit is 3 and 3 is not divisible by 2 | Number is not divisible by 2 |
Answer is 888886 is completely divisible by 2
Divisibility Test for 3 :
If sum of all digits of the number is divisible by 3 then that number is completely divisible by 3.
Look at below numbers and observe:
Number | Observation [sum of all digits] | Divisible by 3 or not |
123 | Sum of all digits ( 1 + 2 + 3 = 6) 6 is divisible by 3 | Number is divisible by 3 |
396 | Sum of all digits (3 + 9 + 6 = 18) 18 -> Sum of digits (1 + 8 = 9) 9 is divisible by 3 | Number is divisible by 3 |
775 | Sum of all digits (7 + 7 + 5 = 19) 19 -> Sum of digits (1 + 9 = 10) 10 is not divisible by 3 | Number is not divisible by 3 |
8679 | Sum of all digits (8 + 6 + 7 + 9 = 30) 30 -> Sum of digits (3 + 0 = 3) 3 is completely divisible by 3 | Number is divisible by 3 |
Note:
When you do sum of all digits and you get answer which is big number. Then continue doing same process till you get single digit sum
Example: 9999968
Sum of digits (9 + 9 + 9 + 9 + 9 + 6 + 8 = 59 )
So some of students may not be able to conclude by looking at number 59 if it is divisible by 3 or not.
Then repeat sum of digit process till you get single digit number
59 -> 5 + 9 = 14
14 -> 1 + 4 = 5
5 is not divisible by 3.
So number 9999968 is not divisible by 3
Problem 1:
Number 5*985 is completely divisible by 3. What number could be there in place of * ? Solution :
As per divisibility test for 3, if sum of all digits of number is divisible by 3 then that number is also divisible by 3
Let us take sum of all digits of number 5*985
Sum = 5 + 9 + 8 + 5 + * = 27 + *
Digit sum of 27 = 2 + 7 = 9 which is divisible by 3 already
In order to have number 5*985 completely divisible sum has of digits should be also divisible by 3.
As Sum without * is already divisible by 3, we need to have such number in place of * which will be divisible by 3 and 3, 6, 9, 0 these numbers are completely divisible by 3.
So answer is 3, 6, 9, 0
Problem 2:
Number 5*699*6 is completely divisible by 3. There is same number in place of both *. What number could be there in place of *?
A)1 B) 0 C) 7 D) 2
Solution :
As per divisibility test for 3, if sum of all digits of number is divisible by 3 then that number is also divisible by 3
Let us take sum of all digits of number 5*985
Sum = 5 + 6 +9 + 9 + 6 + * + * = 35 + 2*. Sum of 35 -> 3 + 5 =8
To find number in place of * we need to make sure:
8 + 2* should be completely divisibly 3.
If we put 1 at * : 8 + (2*1) = 10 not divisible by 3
If we put 2 at *: 8 + (2*2) = 12 divisible by 3. Like this we need to check for 0 to 9 and then conclude.
However if we have options given it is best way to use options to solve such questions.
Options | Observation [ sum of all digits as per divisibility test of 3] | Answer valid/ Not |
A (* = 1) 5169916 | Sum of all digits ( 5 + 1 + 6 + 9 +9 + 1 + 6 = 37) 37 -> Sum of digits (3 + 7 = 10) 10 is not divisible by 3 | Option A is not the answer |
A (* = 0) 5069906 | Sum of all digits (5 + 0 + 6 + 9 +9 + 0 + 6 = 35) 35 -> Sum of digits (3 + 5 = 8) 8 is not divisible by 3 | Option B is not the answer |
A (* = 7) 5769976 | Sum of all digits (5 + 7 + 6 + 9 +9 + 7 + 6 = 49) 49 -> Sum of digits (4 + 9 = 13) 13 -> Sum of digits (1 + 3 = 4) 4 is not divisible by 3 | Option C is not the answer |
A (* = 2) 5269926 | Sum of all digits (5 + 2 + 6 + 9 +9 + 2 + 6 = 39) 39 -> Sum of digits (3 + 9 = 12) -> 1 + 2 = 3 3 is completely divisible by 3 | Option D is correct answer |
Divisibility Test for 4 :
If last 2 digits part of number is divisible by 4 then that number is completely divisible by 4.
So look at number and just focus on right side last 2 digit number
Number | Observation [ last 2 digits] | Divisible by 4 or not |
1236664 | Rightmost Last 2 digits : 64 64 is divisible by 4 | Number is divisible by 4 |
8888860 | Rightmost Last 2 digits : 60 60 is divisible by 4 | Number is divisible by 4 |
1399999 | Rightmost Last 2 digits : 99 99 is not divisible by 4 | Number is not divisible by 4 |
9376544 | Rightmost Last 2 digits : 44 44 is completely divisible by 4 | Number is divisible by 4 |
9000 | Rightmost Last 2 digits : 00 00 is completely divisible by 4 | Number is divisible by 4 |
Problem 1:
Number 665998*0 is completely divisible by 4. What number can be there in place of *? Solution :
As per divisibility test for 4, if last 2 digit number is divisible by 4 then that number is divisible by 4
Let us try to check all combination and then find required number: Focus last 2 digits only
Possibility in place of * | Last 2 digits ( when value put for *) | Possible answer(Yes/No) |
0 | 00 -> Divisible by 4 | Yes |
1 | 10 -> Not divisible by 4 | No |
2 | 20 -> Divisible by 4 | Yes |
3 | 30 -> Not divisible by 4 | No |
4 | 40 -> Divisible by 4 | Yes |
5 | 50 -> Not divisible by 4 | No |
6 | 60 -> Divisible by 4 | Yes |
7 | 70 -> Not divisible by 4 | No |
8 | 80 -> Divisible by 4 | Yes |
9 | 90 -> Not divisible by 4 | No |
Answer is 0, 2, 4, 6, 8
Note:
Most of the time options are given and we can directly refer option to solve this. Let us do next problem with option:
Problem 2:
Number 6977** is completely divisible by 4. What number could be there in place of * ? A) 7 B) 9 C) 5 D)4
Solution :
Let us put all options values at place of * 1 by 1 and check for divisibility test of 4
Options | Observation [ Last 2 digits as per divisibility test of 4] | Answer valid/ Not |
A (* = 7) 697777 | 77 -> Not Divisible by 4 | Option A is not the answer |
B (* = 9) 697799 | 99 -> Not Divisible by 4 | Option B is not the answer |
C (* = 5) 697755 | 55 -> Not Divisible by 4 | Option C is not the answer |
D (* = 4) 697744 | 44 -> Divisible by 4 | Option D is correct answer |
Divisibility Test for 5 :
If Rightmost digit number is 0 or 5 then number is completely divisible by 5. Look at below numbers and observe:
Number | Observation [right most digit 0 0r 5] | Divisible by 5 or not |
35 | Rightmost Last digit is 5 | Number is divisible by 5 |
70 | Rightmost Last digit is 0 | Number is divisible by 5 |
269 | Rightmost Last digit is 9 | Number is not divisible by 5 |
1000 | Rightmost Last digit is 5 | Number is divisible by 5 |
Problem 1:
Number 3789* is completely divisible by 5. What number could be there in place of * ? Solution :
As per divisibility test for 5, any number whose rightmost digit is 0 or 5 is completely divisible by 5.
So in place of * there could be below values:
0 or 5
Answer is 0, 5
Problem 2:
Which of the following numbers are completely divisible by 5 ?
1236665, 888886, 9999995, 9876543
Solution :
As per divisibility test for 5, any number whose rightmost digit is 0 or 5 is divisible by 5.
So let us look at right most number and decide if number is divisible by 5 or not.
Number | Observation [Rightmost digit 0 or 5] | Divisible by 5 or not |
1236665 | Rightmost Last digit is 5 | Number is divisible by 5 |
888886 | Rightmost Last digit is 6 | Number is not divisible by 5 |
9999995 | Rightmost Last digit is 5 | Number is divisible by 5 |
9876543 | Rightmost Last digit is 3 | Number is not divisible by 5 |
Answer is 1236665, 9999995 are completely divisible by 5
Divisibility Test for 6 :
If number is divisible by 2 and 3 then it is completely divisible by 6
Number | Divisible by 2 | Divisible by 3 | Divisible by 6 or not |
360 | Rightmost digit 0 -> Yes | Sum of digit 3 + 6 = 9 9 is divisible by 3 | Number is divisible 6 |
375 | Rightmost digit 5 -> No | Not required | Number is not divisible 6 |
4800 | Rightmost digit 0 -> Yes | Sum of digit 4 + 8 = 12 12-> 1 + 2 =3 3 is divisible by 3 | Number is divisible 6 |
2436 | Rightmost digit 6 -> Yes | Sum of digits 2 + 4 + 3 + 6 = 15 15 -> 1 + 5 = 6 6 is divisible by 3 | Number is divisible 6 |
Divisibility Test for 8 :
If last 3 digits part of number is divisible by 8 then that number is completely divisible by 8.
So look at number and just focus on right side last 3 digit number
Number | Observation [ last 3 digits] | Divisible by 8 or not |
1236064 | Rightmost Last 3 digits : 064 64 is divisible by 8 | Number is divisible by 8 |
8888160 | Rightmost Last 3 digits : 160 60 is divisible by 8 | Number is divisible by 8 |
1399999 | Rightmost Last 3 digits : 999 99 is not divisible by 8 | Number is not divisible by 8 |
9376144 | Rightmost Last 3 digits : 144 144 is completely divisible by 4 | Number is divisible by 8 |
9000 | Rightmost Last 2 digits : 00 00 is completely divisible by 4 | Number is divisible by 8 |
Problem 1:
Number 665998*0 is completely divisible by 8. What number can be there in place of *? Solution :
As per divisibility test for 8, if last 3 digit number is divisible by 8 then that number is divisible by 8
Let us try to check all combination and then find required number: Focus last 3 digits only
Possibility in place of * | Last 3 digits ( when value put for *) | Possible answer(Yes/No) |
0 | 800 -> Divisible by 8 | Yes |
1 | 810 -> Not divisible by 8 | No |
2 | 820 -> Not divisible by 8 | No |
3 | 830 -> Not divisible by 8 | No |
4 | 840 -> Divisible by 8 | Yes |
5 | 850 -> Not divisible by 8 | No |
6 | 860 -> Not divisible by 8 | No |
7 | 870 -> Not divisible by 8 | No |
8 | 880 -> Divisible by 8 | Yes |
9 | 890 -> Not divisible by 8 | No |
Answer is 0, 4, 8
Note:
Most of the time options are given and we can directly refer option to solve this. Let us do next problem with option:
Problem 2:
Number 69777* is completely divisible by 8. What number could be there in place of * ?
A) 5 B) 4 C) 6 D)0
Solution :
Let us put all options values at place of * 1 by 1 and check for divisibility test of 4
Options | Observation [ Last 3 digits as per divisibility test of 8] | Answer valid/ Not |
A (* = 5) 697775 | 775 -> Not Divisible by 8 | Option A is not the answer |
B (* = 4) 697774 | 774 -> Not Divisible by 8 | Option B is not the answer |
C (* = 6) 697776 | 776 -> Divisible by 8 | Option C is correct answer |
D (* = 0) 697770 | 770 -> Not divisible by 8 | Option D is not the answer |
Once we get answer we don’t need to check next options [just written for reference and understanding]
Divisibility Test for 9 :
If sum of all digits of the number is divisible by 9 then that number is completely divisible by 9.
Look at below numbers and observe:
Number | Observation [sum of all digits] | Divisible by 9 or not |
123 | Sum of all digits ( 1 + 2 + 3 = 6) 6 is not divisible by 9 | Number is divisible by 9 |
396 | Sum of all digits (3 + 9 + 6 = 18) 18 -> Sum of digits (1 + 8 = 9) 9 is completely divisible by 9 | Number is divisible by 9 |
775 | Sum of all digits (7 + 7 + 5 = 19) 19 -> Sum of digits (1 + 9 = 10) 10 is not divisible by 9 | Number is not divisible by 9 |
8676 | Sum of all digits (8 + 6 + 7 + 6 = 27) 27 -> Sum of digits (2 + 7 = 9) 9 is completely divisible by 9 | Number is divisible by 9 |
Problem 1:
Number 5*985 is completely divisible by 9. What number could be there in place of * ? Solution :
As per divisibility test for 9, if sum of all digits of number is divisible by 9 then that number is also divisible by 9
Let us take sum of all digits of number 5*985
Sum = 5 + 9 + 8 + 5 + * = 27 + *
Digit sum of 27 = 2 + 7 = 9 which is divisible by 9 already
In order to have number 5*985 completely divisible sum has of digits should be also divisible by 3.
As Sum without * is already divisible by 9, we need to have such number in place of * which will be divisible by 9.
9, 0 these numbers are completely divisible by 9.
So Answer is 9, 0
Problem 2:
Number 5*699*6 is completely divisible by 9. There is same number in place of both *. What number could be there in place of *?
A)1 B) 0 C) 7 D) 2
Solution :
As per divisibility test for 9, if sum of all digits of number is divisible by 9 then that number is also divisible by 9
Let us take sum of all digits of number 5*699*6
Sum = 5 + 6 +9 + 9 + 6 + * + * = 35 + 2*
Sum of 35 -> 3 + 5 =8
To find number in place of * we need to make sure:
8 + 2* should be completely divisibly 9.
If we put 1 at * : 8 + (2*1) = 10 not divisible by 9
If we put 2 at *: 8 + (2*2) = 12 not divisible by 9
Like this we need to check for 0 to 9 and then conclude.
However if we have options given it is best way to use options to solve such questions.
Options | Observation [ sum of all digits as per divisibility test of 9] | Answer valid/ Not |
A (* = 1) 5169916 | Sum of all digits ( 5 + 1 + 6 + 9 +9 + 1 + 6 = 37) 37 -> Sum of digits (3 + 7 = 10) 10 is not divisible by 9 | Option A is not the answer |
A (* = 0) 5069906 | Sum of all digits (5 + 0 + 6 + 9 +9 + 0 + 6 = 35) 35 -> Sum of digits (3 + 5 = 8) 8 is not divisible by 9 | Option B is not the answer |
A (* = 5) 5569956 | Sum of all digits (5 + 5 + 6 + 9 +9 + 5 + 6 = 45) 45 -> Sum of digits (4 + 5 = 9) 9 is divisible by 9 | Option C is the correct answer |
A (* = 2) 5269926 | Sum of all digits (5 + 2 + 6 + 9 +9 + 2 + 6 = 39) 39 -> Sum of digits (3 + 9 = 12) 12 -> Sum of digits (1 + 2 = 3) 3 is not divisible by 9 | Option D is not the answer |
Divisibility Test for 10:
If Rightmost digit of number is 0 then number is completely divisible by 10. Look at below numbers and observe:
Number | Observation [right most digit 0] | Divisible by 10 or not |
3000 | Rightmost Last digit is 0 | Number is divisible by 10 |
70 | Rightmost Last digit is 0 | Number is divisible by 10 |
269 | Rightmost Last digit is 9 | Number is not divisible by 10 |
1000 | Rightmost Last digit is 0 | Number is divisible by 10 |
Problem 1:
Number 3788869* is completely divisible by 10. What number could be there in place of * ?
Solution :
As per divisibility test for 10, any number whose rightmost digit is 0 is completely divisible by 10.
So in place of * there could be below values:
0
Answer is 0
Problem 2:
Which of the following numbers are completely divisible by 10 ?
12366650, 888886, 99999950, 9876543
Solution :
As per divisibility test for 10, any number whose rightmost digit is 0 is completely divisible by 2.
So let us look at right most number and decide if number is divisible by 10 or not.
Number | Observation [Rightmost digit 0 ] | Divisible by 5 or not |
1236665 | Rightmost Last digit is 5 | Number is not divisible by 5 |
8888860 | Rightmost Last digit is 0 | Number is not divisible by 5 |
99999950 | Rightmost Last digit is 0 | Number is not divisible by 10 |
9876543 | Rightmost Last digit is 3 | Number is not divisible by 10 |
Answer is 8888860, 99999950 is completely divisible by 10
Divisibility Test for 11:
Calculate sum of digits at odd places (1, 3, 5…from left)
Calculate sum of digits at even places (2, 4, 6…from left)
Now get difference of both:
Sum of odd places digits – Sum of even places digits.
Sample Number: 278354
2 | 7 | 8 | 3 | 5 | 4 |
Odd place | Even place | Odd place | Even place | Odd place | Even place |
Odd places sum = 2 + 8 + 5 = 15
Even places sum = 7 + 3 + 4 = 14
Difference = Odd place sum – Even places sum
= 15 – 14 = 1
As 1 is not divisible by 11 number 278354 is also not divisible by 11.
If this difference is divisible by 11 then number is divisible by 11 [irrespective sign of difference]
Let us look at some examples to understand:
Number | Sun of odd places digits | Sum of even places digits | Difference [ Odd places sum – even place sum] | Divisible by 11 or not |
451 | 4 + 1= 5 | 5 | 5 -5 = 0 0 is divisible by 11 | Divisible by 11 |
12133 | 1 + 1 + 3 = 5 | 2 + 3 = 5 | 5 – 5 = 0 0 is divisible by 11 | Divisible by 11 |
3916 | 3 + 1 = 4 | 9 + 6 = 15 | 4 – 15 = -11 -11 is divisible by 11 | Divisible by 11 |
7586 | 7 + 8 = 15 | 5 + 6 = 11 | 15 – 11 = 4 4 is not divisible by 11 | Not divisible by 11 |
Summary of Divisibility Tests
Number | Divisibility test |
2 | Rightmost digit number should be divisible by 2 [ 0 ,2, 4, 6, 8 at rightmost] |
3 | Sum of all digits of the number should be divisible by 3 |
4 | Last 2 digit number should be divisible by 4 |
5 | Rightmost digit number should be 0 or 5 |
6 | Number should be divisible by 2 and 3 |
8 | Last 3 digit number should be divisible by 8 |
9 | Sum of all digits of the number should be divisible by 9 |
10 | Rightmost digit number should be 0 |
11 | Sum of odd places digit – sum of even places digit should be divisible by 11 |