DIVISIBILITY
Top tricks to check divisibility of numbers within seconds without calculator. Divisibility
Test are methods to determine whether the given number is divisible by
given divisor/factor or not without actually performing division.
How to find which number is divisible by a given number? In this post first we will learn some simple rules of divisibility and later we will see some complex rules to check divisibility by 7, 13, 17 & 19.
How to find which number is divisible by a given number? In this post first we will learn some simple rules of divisibility and later we will see some complex rules to check divisibility by 7, 13, 17 & 19.
Rules of Divisibility
1. Divisibility by 2: When the last digit of number is even or 0, then the number is divisible by 2.
For example: 450, 654, 889456, etc.
2. Divisibility by 3: When the sum of the digits of any number is divisible by 3, then the number is divisible by 3.
For example: 450, 654, 889456, etc.
2. Divisibility by 3: When the sum of the digits of any number is divisible by 3, then the number is divisible by 3.
For example: 3546 = 3 + 5 + 4 + 6 = 18 which is divisible by 3, so given number is divisible by 3.
3. Divisibility by 4: When the last two digits of any number is divisible by 4, then the number is divisible by 4.
For example: 22564 is divisible by 4 because last two digits '64' are divisible by 4.
4. Divisibility by 5: Numbers having 0 or 5 at units place are divisible by 5.
For example: 225, 66560 etc
5. Divisibility by 6: When any number is divisible by 2 & 3 both, then the number is divisible by 6.
For example: 41682 are divisible by 2 since last digit is even.
Also 41862 = 4 + 1 + 6 + 8 + 2 = 21 which is divisible by 3.
So given number is divisible by 6.
6. Divisibility by 8: When the last three digits of any number is divisible by 8 or when the last three digits of any number are 0, then the number is divisible by 8.
For example: 56256, 254000 etc
7. Divisibility by 9: When the sum of the digits of any number is divisible by 9, then the number is divisible by 9.
For example: 10854 = 1 + 0 + 8 + 5 + 4 = 18 which is divisible by 9, so given number is divisible by 9.
8. Divisibility by 10: Numbers having 0 at units place are divisible by 10.
For example: 654650, 654000 etc.
9. Divisibility by 11: When the difference between the sum of the digits at even and odd places of any number is either zero or multiple of 11, then the number is divisible by 11.
For example: 531421
Sum of digits at even place = 3 + 4 + 1 = 8
Sum of digits at odd place = 5 +1 + 2 = 8
Difference = 8 - 8 = 0
So given number is divisible by 11.
10. Divisibility by 12: When a number is divisible by 3 & 4 both, then the number is divisible by 12.
11. Divisibility by 14: When any number is divisible by 2 & 7 both, then the number is divisible by 14.
12. Divisibility by 15: When any number is divisible by 3 & 5 both, then the number is divisible by 15.
13. Divisibility by 16: When the last 4 digits of any number are divisible by 16, then the number is divisible by 16.
3. Divisibility by 4: When the last two digits of any number is divisible by 4, then the number is divisible by 4.
For example: 22564 is divisible by 4 because last two digits '64' are divisible by 4.
4. Divisibility by 5: Numbers having 0 or 5 at units place are divisible by 5.
For example: 225, 66560 etc
5. Divisibility by 6: When any number is divisible by 2 & 3 both, then the number is divisible by 6.
For example: 41682 are divisible by 2 since last digit is even.
Also 41862 = 4 + 1 + 6 + 8 + 2 = 21 which is divisible by 3.
So given number is divisible by 6.
6. Divisibility by 8: When the last three digits of any number is divisible by 8 or when the last three digits of any number are 0, then the number is divisible by 8.
For example: 56256, 254000 etc
7. Divisibility by 9: When the sum of the digits of any number is divisible by 9, then the number is divisible by 9.
For example: 10854 = 1 + 0 + 8 + 5 + 4 = 18 which is divisible by 9, so given number is divisible by 9.
8. Divisibility by 10: Numbers having 0 at units place are divisible by 10.
For example: 654650, 654000 etc.
9. Divisibility by 11: When the difference between the sum of the digits at even and odd places of any number is either zero or multiple of 11, then the number is divisible by 11.
For example: 531421
Sum of digits at even place = 3 + 4 + 1 = 8
Sum of digits at odd place = 5 +1 + 2 = 8
Difference = 8 - 8 = 0
So given number is divisible by 11.
10. Divisibility by 12: When a number is divisible by 3 & 4 both, then the number is divisible by 12.
11. Divisibility by 14: When any number is divisible by 2 & 7 both, then the number is divisible by 14.
12. Divisibility by 15: When any number is divisible by 3 & 5 both, then the number is divisible by 15.
13. Divisibility by 16: When the last 4 digits of any number are divisible by 16, then the number is divisible by 16.
OSCULATOR
To
understand the divisibility by 7, 13, 17, 19 or any other number which
is either '1' more or less than any multiply of '10' or any number whose
multiple is either '1' more or less than '10' we need to understand the
meaning of Osculator.
There are two types of Osculator:
1. One-more or positive Osculator: It means the number needs 1 more to be a multiple of 10.
For example: 19 needs 1 to become a multiple of 10, i.e 19 + 1 = 20 (= 2 x 10). Thus One-more Osculator of 19 is '2'.
2. One-less or negative Osculator: It means the number needs to be reduced by 1 to be a multiple of 10.
For example: 21 need to be reduced by 1 to become a multiple of 10, i.e 21 - 1 = 20 (= 2 x 10). Thus Negative Osculator of 21 is '2'.
Finding the Osculator of 7, 13, 17 & 19:
There are two types of Osculator:
1. One-more or positive Osculator: It means the number needs 1 more to be a multiple of 10.
For example: 19 needs 1 to become a multiple of 10, i.e 19 + 1 = 20 (= 2 x 10). Thus One-more Osculator of 19 is '2'.
2. One-less or negative Osculator: It means the number needs to be reduced by 1 to be a multiple of 10.
For example: 21 need to be reduced by 1 to become a multiple of 10, i.e 21 - 1 = 20 (= 2 x 10). Thus Negative Osculator of 21 is '2'.
Finding the Osculator of 7, 13, 17 & 19:
- Osculator of 7: Look for that multiple of 7 which is either 1 more or less than any multiple of 10.
7 x 3 = 21 has Negative Osculator which is '2'. Also 7 x 7 = 49 has One-more Osculator which is '5'.
- Osculator of 13: 13 x 3 = 39, so One-more Osculator of 13 is '4'.
- Osculator of 17: 17 x 3 = 51, so Negative Osculator of 17 is '5'.
- Osculator of 19: 19 x 1 = 19, so One-more Osculator of 19 is '2'.
14. Divisibility by 7: We know that '2' is the Negative Osculator of 7, so multiply the last digit of any number by '2' and subtract this product from rest of the number. Repeat this step and check if the remainder is divisible by 7. If the remainder is divisible by 7 then the number is also divisible by 7.
Example: Is 2961 divisible by 7?
Solution: We need to multiply the last digit with 2 and subtract from the remaining digits.
⇒ 296 - 1 x 2 = 294
⇒ 29 - 4 x 2 = 21 which is divisible by 7.
It means given number is also divisible by 7.
Second Method: For large numbers, Forming alternating sum of blocks of 3 digits from right to left with positive and negative sign respectively gives us multiple of 7.
Example: Is 55277838 divisible by 7?
Solution: 838 - 277 + 55 = 616 = 7 x 88
or we can check 616 = 61 - 6 x 2 = 49 which is a multiple of 7.
So given number is divisible by 7.
15. Divisibility by 13: We know that '4' is the One-more Osculator of 13, so multiply the last digit of any number by '4' and add this product to the rest of the number. Check if the result is divisible by 13 or not. If it is divisible by 13 then the number is divisible by 13.
Example: Is 4212 divisible by 13?
Solution: We need to multiply the last digit with 4 and add to the remaining digits.
Clearly 78 is divisible by 13, so given number is divisible by 13.
Solution: We need to multiply the last digit with 4 and add to the remaining digits.
Clearly 78 is divisible by 13, so given number is divisible by 13.
Second Method:
For large numbers, form alternating sum of blocks of 3 digits from
right to left with positive and negative sign respectively, if the
result is divisible by 13 then number will be divisible by 13.
Example: Is 24167 divisible by 13?
Solution: 167 - 24 = 143 (= 13 x 11)
or we can check 143 = 14 + 3 x 4 = 26 which is a multiple of 13.
So given number is divisible by 7.
16. Divisibility by 17: Negative osculator of 17 is '5', so multiply last digit with 5 and repeat same steps of divisibility by 7.
17. Divisibility by 19: One-more Osculator for 19 is '2', so multiply last digit of any number by 2 and add to the remaining number. Steps are same as divisibility by 13.
Second Method: Add 4 times the last two digits to the rest of number, if the result is divisible by 19 then the number is divisible by 19.
18. Divisibility by 25: When the last two digits of number are either 0 or divisible by 25, then the number will be divisible by 25.
19. Any single digit number written n-1 times will be divisible by n, where n is a prime number > 5
Example: 333333333333 is divisible by 13
These are some best divisibility test methods. No Rule needs pen & paper, the above calculation can be done in one line or even mentally to find if the given number is divisible by certain number or not. By adopting these methods you can check the divisibility of almost every number. These methods are useful in simplifying calculation especially questions of simplifications, remainder etc. Good Luck 👍
Example: Is 24167 divisible by 13?
Solution: 167 - 24 = 143 (= 13 x 11)
or we can check 143 = 14 + 3 x 4 = 26 which is a multiple of 13.
So given number is divisible by 7.
16. Divisibility by 17: Negative osculator of 17 is '5', so multiply last digit with 5 and repeat same steps of divisibility by 7.
17. Divisibility by 19: One-more Osculator for 19 is '2', so multiply last digit of any number by 2 and add to the remaining number. Steps are same as divisibility by 13.
Second Method: Add 4 times the last two digits to the rest of number, if the result is divisible by 19 then the number is divisible by 19.
18. Divisibility by 25: When the last two digits of number are either 0 or divisible by 25, then the number will be divisible by 25.
19. Any single digit number written n-1 times will be divisible by n, where n is a prime number > 5
Example: 333333333333 is divisible by 13
These are some best divisibility test methods. No Rule needs pen & paper, the above calculation can be done in one line or even mentally to find if the given number is divisible by certain number or not. By adopting these methods you can check the divisibility of almost every number. These methods are useful in simplifying calculation especially questions of simplifications, remainder etc. Good Luck 👍
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