# Divisibility Rules/Tests In Math |From 1 to 14.

Divisibility Rules/Tests In Math

## 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.

## FAQs

Question 1. What is meant by divisibility rules?

Answer. Divisibility Rules: The rules which are used to test whether a number is completely divisible by another number or not without any calculation especially the division.

Question 2.What is the divisibility rule for 2 and 5?

Answer. Divisibility rules for 2: A number is divisible by 2 , if the last digit of that number is even (0, 2, 4, 6, 8).

Divisibility rule for 5:  If the unit’s digit of a number is either 0 or 5, then the number is divisible by 5.

Question 3.What is the divisibility rule for 7 and give an example?

Answer.  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.

Question 4. What is the divisibility rule for 13?

Answer:    Calculate the four times 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.

 Example:    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.

Question 5. What is the divisibility rule of 7 and 11?

Divisibility rule of 7Double 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.

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

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

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 numberif 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.

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.

Divisibility rule for 2If 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 3Sum 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?

AnswerFor that we should know the divisibility rule of 4.

### Here  given number is 8614_2.

Now we will find the multiples of 4.

List of multiples of 4-

4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92.

From this above list we will take only two digit numbers with last digit 2.

So here is the list of numbers- 12,32,52,72,92.

### Question 6:If a 3 digit number 61y is divisible by 3 then write all the possible values of y.

Answer-We know the divisibility tests for 3.

if the sum of the digits available in number is divisible by 3 then the number is divisible by 3.
given number is =61y.
sum of digits=6+1+y=7+y.
Multiples of 3 are =6,9,12,15,18,21…..
•        7+y=9
•        7+y=12
•        7+y=15
we can consider only 9,12,15 among the list of multiples of 3.

Then  Possible values for y
•     y=2     (y=9-7),
•     y=5      (y=12-7),
•     y=8      (y=15-7).
For the possible values of y ,the number will be  612,615,618.

### Question 7: Find out if 463 is divisible by 2?

Answer-  First consider divisibility rules for 2.
It shows if last digit of a number is even (0, 2, 4, 6, 8),then given number is   divisible by 2.
Here unit digit is 3.
So the number 463 is not divisible by 2.