Divisibility Rules/Tests In Math
Table of Contents
Divisibility Rules
Divisibility Rules/Tests In Math are the rules which let us test if one number can be completely divided by another without too much calculation especially division. Divisibility rules/Tests In Math are shorthand rules of knowing whether a number is divisible by a divisor without performing the division.
|
Question 5. What is the divisibility rule of 7 and 11? Answer: Divisibility rule of 7 – Double the last/unit digit and subtract it from the rest of the number and if the result is 0 or divisible by 7 then number is divisible by 7. Example: Ex-672 (double of 2 is 4 and 67-4=63 and 63 is divisible by 7.
Divisibility rule of 11- A given number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at the even places is either 0 or a number divisible by 11. |
Before we understand the divisibility rules we should know what is divisibility?
Divisibility :-
The capacity of being divided. Especially with no remainder leftover. In math, we can say a number is completely divisible, If a number (dividend) is divided by a another number(divisor) and gives a whole number (quotient) as a result with 0 as remainder .
Divisibility Rules/Tests:
The rules which are used to test whether a number is completely divisible by another number without any calculation especially the division.
 |
| Divisibility Rules In Math |
The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest.
Divisibility rules/Tests from 1 to 14.
|
divisor |
Divisibility condition |
examples |
Divisibility rule for 1 |
No special condition. Any integer is divisible by 1 but 1 is not divisible by any number. |
13 is divisible by 1 and 987677 is also divisible by 1.no matter what a big number is. |
Divisibility rule for 2 |
Last digit of a number is even (0, 2, 4, 6, 8). |
35642856: 6 is even.
|
Divisibility rule for 3 |
Sum of the digits must be divisible by 3.
|
357—3+8+7=15 and it is divisible by 3.
|
Divisibility rule for 4 |
A number is divisible by 4 if the number formed by the last two digits is divisible by 4. |
Ex-56524 is divisible by 4, since last two digits 24 is divisible by 4.
45653 is not divisible by 4, since last two digit 53 is not divisible by 4.
|
Divisibility rule for 5 |
If the unit’s digit of a number is either 0 or 5, then the number is divisible by 5.
|
Each one of the numbers23540,23455 is divisible by 5 since, they end is 0 and 5 respectively.
56433 is not divisible by 5 since, unit digit is 3 not 0 or 5.
|
Divisibility rule for 6 |
A number is divisible by 6 if
(i) It is an even number and
(ii) It is divisible by 3.
|
126 .it is even and 1+2+6=9 and 9 is divisible by 3.so 126 is divisible by 6.
|
Divisibility rule for 7 |
Double the last/unit digit and subtract it from the rest of the number and if the result is 0 or divisible by 7 then number is divisible by 7.
|
Ex-672 (double of 2 is 4 and 67-4=63 and 63 is divisible by 7.
one more example-
126 (6*2=12 and 12-12=0. The result is 0, so the number is divisible by 7.)
|
Divisibility rule for 8 |
A given number is divisible by 8, if the number formed by the last 3 digits of the given number are either 000 or divisible by 8.
|
If we consider 6774512, then the number formed by last 3 digits is 512, which is divisible by 8. Hence the given number is divisible by 8. 67000 ,which is divisible by 8 as its last three digits are 000
|
Divisibility rule for 9 |
A number is divisible by 9, if the sum of the digits of the number is divisible by 9.
|
Ex-if we consider the number 34533, which is divisible by 9. Hence given number is divisible by 9.
|
Divisibility rule for 10 |
A number is divisible by 10 if the unit digit of the number is 0.
|
Ex – 242430. Unit digit is 0 so number is divisible by 10.
|
Divisibility rule for 11 |
A given number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at the even places is either 0 or a number divisible by 11.
|
Ex- consider the number 1364.
Sum of the digits at odd places number is=1+6=7.
Sum of the digits at even places number is=3+4=7.
Difference between sum of the digits at odd places and sum of the digits at even places is
= 7-7=0.
Hence given number is divisible by 11.
|
Divisibility rule for 12 |
A number is divisible by 12 if given number is divisible by both 3 and 4.
|
Ex- 324.
324 is divisible by 3 (3+2+4=9 and 9 is divisible by 3.) and 324 is also divisible by 4 (last two digits 24 is divisible 4). Hence 324 is divisible by 12.
|
Divisibility rule for 13 |
Calculate the four times of the unit digit of the number and add it to the remaining leading number. If result is divisible by 13 then the given number is divisible by 13. |
Ex- 325 last digit is 5 and four times of 5 is =5*4=20 add it to the remaining leading number 32+20=52 which is divisible by 13. or we can take one more step for easy calculation. again consider 52.four times of 2=8 add 5 and 8. 5+8=13. so 325 is divisible by 13. |
Divisibility rule for 14 |
A number is divisible by 14,if the number is divisible by 2 and by 7. |
Ex- 294 294 is divisible 2 because its last digit is 4. 294 is divisible by 7 as (two times of last digit is 8.and 29-8=21 which is divisible by 7). Hence 294 is divisible by 14.
|
Divisibility Rules/Tests for 17:
Subtract five times of the unit digit from remaining number, if the result is 0
or divisible by 17 then the number is divisible by 17.
Example- check 357 is divisible by 17 or not.
· Calculate five times of the unit digit of given number i.e. 7*5=35.
· Subtract above multiplication value from remaining numer.i.e.35-35=0
· If result is 0 or divisible by 17 then the given number is divisible by 17.Here ‘0’ came as result. Hence, 357 is divisible by 17.
Divisibility Rules/Tests for 19:
Add two times the unit/last digit of the number to the rest of the number and if the result is divisible by 19 then the given number is divisible by 19.
Example: – check 437 is divisible by 19 or not?
o Last digit of the number is =7
o Double the last digit means 7*2=14
o Now add 14 to the rest of the number 43. Result=43+14=57.
Which is divisible by 19.
Related topics to read
Questions and Answers
Question 1. 675348p32 is divisible by 3.Find the all possible values of 3?
Answer- According to divisibility rule for 3, if the sum of the digits available in number is divisible by 3 then the number is divisible by 3.
Here 6+7+5+3+4+8+3+2=38.which is not divisible by 3.
For divisibility, we need 1, 4 and 7.
So 1,4 and 7 are the possible values of p.
Question 2:- Check divisibility rule for 11 for 10934.
Divisibility rule for 11
A number is divisible by 11 if the difference between sum of the digits at odd places and sum of the digits at the even places is either 0 or a number divisible by 11.
Consider 10934.
· Sum of digits at odd places = 1+9+4=14.
· Sum of digits at even places = 0+3=3.
· Difference = 14-3=11 which is divisible by 11.
Q Question 3:- Test the divisibility of 380 by 2 3 and 5.
Answer:-
Divisibility rule for 2 –If last digit of a number is even (0, 2, 4, 6, and 8), then the number is divisible by 2.
Here considered the number 380 .Its last digit is 0.so number is divisible by 2.
Divisibility rule for 3 – Sum of the digits must be divisible by 3.
Sum of the digits= 3+8+0=11.which is not divisible by 3.so 380 is not divisible by 3.
Divisibility rule for 5 – If the unit’s digit of a number is either 0 or 5, then the number is divisible by 5.
Given number 380.Its last digit is 0.so 380 is divisible by 5.
So our last Answer is 380 is divisible by 2 and 5 but not divisible by 3.
Question 4: What least value should be given to *so that the number 8456 *4107 is divisible by 3?
Answer-Divisibility rule of 3 – Sum of the digits must be divisible by 3.
In this given number (8456 *4107) sum of the digits is
=8+4+5+6+4+1+0+7
=35.
which is not divisible by 3.
so we can not put * as 0.
but we can take * as 1.
If we will put * as 1 the sum would be =8+4+5+6+1+4+1+0+7
=36(again 3+6=9).
which is divisible by 3.
Hence 1 is the least value that can give to * so that the number
8456 *4107 is divisible by 3.
Question 5:How to make 8614_2 divisible by 4?
Answer–For that we should know the divisibility rule of 4.
Divisibility rule for 4:– A number is divisible by 4 if the number formed by last two digits of the given number is divisible by 4.
Here given number is 8614_2.
Answer is we can put 1,3,5,7,9 in place of _ in the given number 8614_2.
Question 6:If a 3 digit number 61y is divisible by 3 then write all the possible values of y.
- 7+y=9
- 7+y=12
- 7+y=15
- y=2 (y=9-7),
- y=5 (y=12-7),
- y=8 (y=15-7).
Question 7: Find out if 463 is divisible by 2?