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# Important Formulas(Part 2) - Numbers

## Divisible by

One whole number is divisible by another if the remainder we get after the division is zero.

### Examples

• $36$ is divisible by $4$ because $36÷4=9$ with a remainder of $0$
• $36$ is divisible by $6$ because $36÷6=6$ with a remainder of $0$
• $36$ is not divisible by $5$ because $36÷5=7$ with a remainder of $1$

## Divisibility Rules

Using divisibility rules, we can easily find whether a given number is divisible by another number without actually performing the division. This saves time especially when working with numbers. Divisibility rules of numbers $1$ to $20$ are provided below.

## Divisibility by 2

A number is divisible by $2$ if the last digit is even, i.e., if the last digit is $0,2,4,6\text{ or }8$

### Example 1

Check if $64$ is divisible by $2$
The last digit of $64$ is $4$ (even).
Hence $64$ is divisible by $2$

### Example 2

Check if $69$ is divisible by $2$
The last digit of $69$ is $9 ($not even).
Hence $69$ is not divisible by $2$

## Divisibility by 3

A number is divisible by $3$ if the sum of the digits is divisible by $3$

Please note that we can apply this rule to the answer again and again if needed.

### Example 1

Check if $387$ is divisible by $3$
$3+8+7=18.$
$18$ is divisible by $3$
Hence $387$ is divisible by $3$

### Example 2

Check if $421$ is divisible by $3$
$4+2+1=7.$
$7$ is not divisible by $3$
Hence $421$ is not divisible by $3$

## Divisibility by 4

A number is divisible by $4$ if the number formed by the last two digits is divisible by $4$

### Example 1

Check if $416$ is divisible by $4$
Number formed by the last two digits $=16$
$16$ is divisible by $4$
Hence $416$ is divisible by $4$

### Example 2

Check if $481$ is divisible by $4$
Number formed by the last two digits $=81$
$81$ is not divisible by $4$
Hence $481$ is not divisible by $4$

## Divisibility by 5

A number is divisible by $5$ if the last digit is either $0$ or $5$

### Example 1

Check if $305$ is divisible by $5$
Last digit is $5.$
Hence $305$ is divisible by $5$

### Example 2

Check if $420$ is divisible by $5$
Last digit is $0.$
Hence $420$ is divisible by $5$

### Example 3

Check if $312$ is divisible by $5$
Last digit is $2.$
Hence $312$ is not divisible by $5$

## Divisibility by 6

A number is divisible by $6$ if it is divisible by both $2$ and $3$

### Example 1

Check if $546$ is divisible by $6$
$546$ is divisible by $2$
$546$ is also divisible by $3$ (Refer divisibility rule of $2$ and $3$)
Hence $546$ is divisible by $6$

### Example 2

Check if $633$ is divisible by $6$
$633$ is not divisible by $2$ though it is divisible by $3$
Hence $633$ is not divisible by $6$

### Example 3

Check if $635$ is divisible by $6$
$635$ is not divisible by $2$
$635$ is also not divisible by $3$
Hence $635$ is not divisible by $6$

### Example 4

Check if $428$ is divisible by $6$
$428$ is divisible by $2$ but it is not divisible by $3$
Hence $428$ is not divisible by $6$

## Divisibility by 7

To find out if a number is divisible by $7,$ double the last digit and subtract it from the number formed by the remaining digits.

Repeat this process until we get at a smaller number whose divisibility we know.

If this smaller number is $0$ or divisible by $7,$ the original number is also divisible by $7$

### Example 1

Check if $349$ is divisible by $7$
Given number $=349$
$34-(9×2)=34-18=16$
$16$ is not divisible by $7$
Hence $349$ is not divisible by $7$

### Example 2

Check if $364$ is divisible by $7$
Given number $=364$
$36-(4×2)=36-8=28$
$28$ is divisible by $7$
Hence $364$ is also divisible by $7$

### Example 3

Check if $3374$ is divisible by $7$
Given number $=3374$
$337-(4×2)=337-8=329$
$32-(9×2)=32-18=14$
$14$ is divisible by $7$
Hence $329$ is also divisible by $7$
Hence $3374$ is also divisible by $7$

## Divisibility by 8

A number is divisible by $8$ if the number formed by the last three digits is divisible by $8$

### Example 1

Check if $7624$ is divisible by $8$
The number formed by the last three digits of $7624=624.$
$624$ is divisible by $8$
Hence $7624$ is also divisible by $8$

### Example 2

Check if $129437464$ is divisible by $8$
The number formed by the last three digits of $129437464=464.$
$464$ is divisible by $8$
Hence $129437464$ is also divisible by $8$

### Example 3

Check if $737460$ is divisible by $8$
The number formed by the last three digits of $737460=460.$
$460$ is not divisible by $8$
Hence $737460$ is also not divisible by $8$

## Divisibility by 9

A number is divisible by $9$ if the sum of its digits is divisible by $9$

(Please note that we can apply this rule to the answer again and again if we need)

### Example 1

Check if $367821$ is divisible by $9$
$3+6+7+8+2+1=27$
$27$ is divisible by $9$
Hence $367821$ is also divisible by $9$

### Example 2

Check if $47128$ is divisible by $9$
$4+7+1+2+8=22$
$22$ is not divisible by $9$
Hence $47128$ is not divisible by $9$

### Example 3

Check if $4975291989$ is divisible by $9$
$4+9+7+5+2+9+1+9+8+9=63$
Since $63$ is big, we can use the same method to see if it is divisible by $9$
$6+3=9$
$9$ is divisible by $9$
Hence $63$ is also divisible by $9$
Hence $4975291989$ is also divisible by $9$

## Divisibility by 10

A number is divisible by $10$ if the last digit is $0.$

### Example 1

Check if $2570$ is divisible by $10$
Last digit is $0.$
Hence $2570$ is divisible by $10$

### Example 2

Check if $5462$ is divisible by $10$
Last digit is not $0.$
Hence $5462$ is not divisible by $10$

## Divisibility by 11

To find out if a number is divisible by $11,$ find the sum of the odd numbered digits and the sum of the even numbered digits.

Now subtract the lower number obtained from the bigger number obtained.

If the number we get is $0$ or divisible by $11,$ the original number is also divisible by $11$

### Example 1

Check if $85136$ is divisible by $11$
$8+1+6=15$
$5+3=8$
$15-8=7$
$7$ is not divisible by $11$
Hence $85136$ is not divisible by $11$

### Example 2

Check if $2737152$ is divisible by $11$
$2+3+1+2=8$
$7+7+5=19$
$19-8=11$
$11$ is divisible by $11$
Hence $2737152$ is also divisible by $11$

### Example 3

Check if $957$ is divisible by $11$
$9+7=16$
$5=5$
$16-5=11$
$11$ is divisible by $11$
Hence $957$ is also divisible by $11$

### Example 4

Check if $9548$ is divisible by $11$
$9+4=13$
$5+8=13$
$13-13=0$
We got the difference as $0.$
Hence $9548$ is divisible by $11$

## Divisibility by 12

A number is divisible by $12$ if the number is divisible by both $3$ and $4$

### Example 1

Check if $720$ is divisible by $12$
$720$ is divisible by $3$
$720$ is also divisible by $4$ (Refer divisibility rules of $3$ and $4$)
Hence $720$ is divisible by $12$

### Example 2

Check if $916$ is divisible by $12$
$916$ is not divisible by $3 ,$ though $916$ is divisible by $4$
Hence $916$ is not divisible by $12$

### Example 3

Check if $921$ is divisible by $12$
$921$ is divisible by $3$
But $921$ is not divisible by $4$
Hence $921$ is not divisible by $12$

### Example 4

Check if $827$ is divisible by $12$
$827$ is not divisible by $3. 827$ is also not divisible by $4$
Hence $827$ is not divisible by $12$

## Divisibility by 13

To find out if a number is divisible by $13,$ multiply the last digit by $4$ and add it to the number formed by the remaining digits.

Repeat this process until we get at a smaller number whose divisibility we know.

If this smaller number is divisible by $13,$ the original number is also divisible by $13$

### Example 1

Check if $349$ is divisible by $13$
Given number $=349$
$34+(9×4)=34+36=70$
$70$ is not divisible by $13$
Hence $349$ is not divisible by $349$

### Example 2

Check if $572$ is divisible by $13$
Given number $=572$
$57+(2×4)=57+8=65$
$65$ is divisible by $13$
Hence $572$ is also divisible by $13$

### Example 3

Check if $68172$ is divisible by $13$
Given number $=68172$
$6817+(2×4)=6817+8=6825$
$682+(5×4)=682+20=702$
$70+(2×4)=70+8=78$
$78$ is divisible by $13$
Hence $68172$ is also divisible by $13$

### Example 4

Check if $651$ is divisible by $13$
Given number $=651$
$65+(1×4)=65+4=69$
$69$ is not divisible by $13$
Hence $651$ is not divisible by $13$

## Divisibility by 14

A number is divisible by $14$ if it is divisible by both $2$ and $7.$

### Example 1

Check if $238$ is divisible by $14$
$238$ is divisible by $2$
$238$ is also divisible by $7$ (Refer divisibility rule of $2$ and $7$)
Hence $238$ is divisible by $14$

### Example 2

Check if $336$ is divisible by $14$
$336$ is divisible by $2$
$336$ is also divisible by $7$
Hence $336$ is divisible by $14$

### Example 3

Check if $342$ is divisible by $14$
$342$ is divisible by $2$
But $342$ is not divisible by $7$
Hence $342$ is not divisible by $12$

### Example 4

Check if $175$ is divisible by $14$
$175$ is not divisible by $2,$ though it is divisible by $7$
Hence $175$ is not divisible by $14$

### Example 5

Check if $337$ is divisible by $14$
$337$ is neither divisible by $2$ nor by $7$
Hence $337$ is not divisible by $14$

## Divisibility by 15

A number is divisible by $15$ if it is divisible by both $3$ and $5.$

### Example 1

Check if $435$ is divisible by $15$
$435$ is divisible by $3$
$435$ is also divisible by $5$ (Refer divisibility rule of $3$ and $5$)
Hence $435$ is divisible by $15$

### Example 2

Check if $555$ is divisible by $15$
$555$ is divisible by $3$
$555$ is also divisible by $5$
Hence $555$ is also divisible by $15$

### Example 3

Check if $483$ is divisible by $15$
$483$ is divisible by $3$
But $483$ is not divisible by $5$
Hence $483$ is not divisible by $15$

### Example 4

Check if $485$ is divisible by $15$
$485$ is not divisible by $3,$ though it is divisible by $5$
Hence $485$ is not divisible by $15$

### Example 5

Check if $487$ is divisible by $15$
$487$ is not divisible by $3$
It is also not divisible by $5$
Hence $487$ is not divisible by $15$

## Divisibility by 16

A number is divisible by $16$ if the number formed by the last four digits is divisible by $16$

### Example 1

Check if $5696512$ is divisible by $16$
The number formed by the last four digits of $5696512=6512$
$6512$ is divisible by $16$
Hence $5696512$ is also divisible by $16$

### Example 2

Check if $3326976$ is divisible by $16$
The number formed by the last four digits of $3326976=6976$
$6976$ is divisible by $16$
Hence $3326976$ is also divisible by $16$

### Example 3

Check if $732374360$ is divisible by $16$
The number formed by the last three digits of $732374360=4360$
$4360$ is not divisible by $16$
Hence $732374360$ is also not divisible by $16$

## Divisibility by 17

To find out if a number is divisible by $17,$ multiply the last digit by $5$ and subtract it from the number formed by the remaining digits.

Repeat this process until you arrive at a smaller number whose divisibility you know.

If this smaller number is divisible by $17,$ the original number is also divisible by $17$

### Example 1

Check if $500327$ is divisible by $17$
Given Number $=500327$
$50032-(7×5 )=50032-35=49997$
$4999-(7×5 )=4999-35=4964$
$496-(4×5 )=496-20=476$
$47-(6×5 )=47-30=17$
$17$ is divisible by $17$
Hence $500327$ is also divisible by $17$

### Example 2

Check if $521461$ is divisible by $17$
Given Number $=521461$
$52146-(1×5 )=52146-5=52141$
$5214-(1×5 )=5214-5=5209$
$520-(9×5 )=520-45=475$
$47-(5×5 )=47-25=22$
$22$ is not divisible by $17$
Hence $521461$ is not divisible by $17$

## Divisibility by 18

A number is divisible by $18$ if it is divisible by both $2$ and $9.$

### Example 1

Check if $31104$ is divisible by $18$
$31104$ is divisible by $2$
$31104$ is also divisible by $9$ (Refer divisibility rule of $2$ and $9$)
Hence $31104$ is divisible by $18$

### Example 2

Check if $1170$ is divisible by $18$
$1170$ is divisible by $2$
$1170$ is also divisible by $9$
Hence $1170$ is divisible by $18$

### Example 3

Check if $1182$ is divisible by $18$
$1182$ is divisible by $2$
But $1182$ is not divisible by $9$
Hence $1182$ is not divisible by $18$

### Example 4

Check if $1287$ is divisible by $18$
$1287$ is not divisible by $2$ though it is divisible by $9$
Hence $1287$ is not divisible by $18$

## Divisibility by 19

To find out if a number is divisible by $19,$ multiply the last digit by $2$ and add it to the number formed by the remaining digits.

Repeat this process until you arrive at a smaller number whose divisibility you know.

If this smaller number is divisible by $19,$ the original number is also divisible by $19$

### Example 1

Check if $74689$ is divisible by $19$
Given Number $=74689$
$7468+(9×2 )=7468+18=7486$
$748+(6×2 )=748+12=760$
$76+(0×2 )=76+0=76$
$76$ is divisible by $19$
Hence $74689$ is also divisible by $19$

### Example 2

Check if $71234$ is divisible by $19$
Given Number $=71234$
$7123+(4×2 )=7123+8=7131$
$713+(1×2 )=713+2=715$
$71+(5×2 )=71+10=81$
$81$ is not divisible by $19$
Hence $71234$ is not divisible by $19$

## Divisibility by 20

A number is divisible by $20$ if it is divisible by $10$ and the tens digit is even.

(There is one more rule to see if a number is divisible by $20$ which is given below.
A number is divisible by $20$ if the number is divisible by both $4$ and $5)$

### Example 1

Check if $720$ is divisible by $20$
$720$ is divisible by $10$ (Refer divisibility rule of $10$)
The tens digit $=2=$ even digit.
Hence $720$ is also divisible by $20$

### Example 2

Check if $1340$ is divisible by $20$
$1340$ is divisible by $10$
The tens digit $=4=$ even digit.
Hence $1340$ is divisible by $20$

### Example 3

Check if $1350$ is divisible by $20$
$1350$ is divisible by $10$
But the tens digit $=5=$ not an even digit.
Hence $1350$ is not divisible by $20$

### Example 4

Check if $1325$ is divisible by $20$
$1325$ is not divisible by $10$ though the tens digit $=2=$ even digit.
Hence $1325$ is not divisible by $20$

## Practise Divisibility Rules

You can use divisibility calculator to find divisibility using the rules mentioned above.