Rules of divisibility part -1

Rules of divisibility

In Vedic mathematics, it can be determined whether any part of any other number can be given completely without dividing it by several methods. These methods are based on the law of divisibility. By using these rules, calculations related to factors, parts etc. are simplified.

Law of divisibility by number 2 :- If unit digit of given number is divisible by 2 then the given number is divisible by 2 also.

                                                                or 

If unit digit of given number is 0,2,4,6 or 8 , then given number is divisible by 2.

Example :- 10 ,32 ,74 ,108 ,2058 etc.

Law of divisibility by number 3 :-If the sum of digits of given number is divisible by 3, then the number is divisible by 3 also.

Illustration 1:- In 546532

sum of digits=5+4+6+5+3+2=25

25 is not divisible by 3, Hence the number 546532 is not divisible by 3.

Illustration 2:- In 287286

sum of digits=2+8+7+2+8+6=33

33 is divisible by 3, Hence the number 287286 is divisible by 3.

Law of divisibility by number 4 :-If the number made up of unit and tens digit is divided by 4, then the whole number is divided by 4.

                                                                    or 

If the number made up of last two digits of given number is divided by 4, then the whole number is divided by 4.

Illustration 1:- In 6532,  the number made up of last two digits is 32. Which is divisible by 4, Hence the number 6532 is also divisible by 4.

Illustration 2:- In 7286,  the number made up of last two digits is 86. Which is not divisible by 4, Hence the number 7286 is not divisible by 4.

Illustration 3:- In 78688,  the number made up of last two digits is 88. Which is divisible by 4, Hence the number 78688 is also divisible by 4.

Law of divisibility by number 5 :-If the last digit of given number is 0 or 5, then the number is divisible by 5.

Illustration 1:-In 585

last digit is 5, hence 585 is divisible by 5.

Illustration 2:-In 56580

last digit is 0, hence 56580 is divisible by 5.

Illustration 1:-In 5856

last digit is not 0 or 5, hence 5856 is not divisible by 5. 

Law of divisibility by number 6 :-If given number is divisible by 2 and 3 both, then the number is divisible by 6 also.

Illustration 1:- In 186

last digit is 6, so 186 is divisible by 2 and 

sum of digits=1+8+6=15

15 is divisible by 3, so the number 186 is divisible by 3.

Hence the number 186 is divisible by 6 also.

(because 186 is divisible by 2 and 3 both)

Illustration 2:- In 585

last digit is 5, so 185 is not divisible by 2 and 

sum of digits=5+8+5=18

18 is divisible by 3, so the number 585 is divisible by 3.

Hence the number 585 is not divisible by 6.

(because 585 is divisible by 3 but not divisible by 2)

Illustration 3:- In 256

last digit is 6, so 256 is divisible by 2 and 

sum of digits=2+5+6=13

13 is not divisible by 3, so the number 256 is not divisible by 3.

Hence the number 256 is not divisible by 6.

(because 256 is divisible by 2 but not divisible by 3)

Law of divisibility by number 7 :-From the right side of the number (by unit digit), make groups of 3–3 digits. If there are more than 2 groups, then add them according to the even or odd place.

         If the difference of their sums is divisible by 7, then the whole number will be divisible by 7.

Illustration 1:-In 315724582

From right side groups are 315 , 724 , 582

First group =582+315=897

Second group=724

difference=897-724=173

173 is not divisible by 7, Hence 315724582 is not divisible by 7.

Illustration 2:-In 22386

From right side groups are 22 , 386

First group =386

Second group=22

difference=386-22=364

364 is divisible by 7, Hence 22386 is divisible by 7.

Law of divisibility by number 8 :-If the number made up of unit, tens and hundred digit is divided by 8, then the whole number is divided by 8.

                                                                    or 

If the number made up of last three digits of given number is divided by 8, then the whole number is divided by 8.

Illustration 1:- In 6532,  the number made up of last three digits is 532. Which is not  divisible by 8, Hence the number 6532 is not divisible by 8.

Illustration 2:- In 7286,  the number made up of last three digits is 286. Which is not divisible by 8, Hence the number 7286 is not divisible by 8.

Illustration 3:- In 78688,  the number made up of last three digits is 688. Which is divisible by 8, Hence the number 78688 is also divisible by 8.

Law of divisibility by number 9 :-If the sum of digits of given number is divisible by 9, then the given number is divisible by 9 also.

Illustration 1:- In 5426532

sum of digits=5+4+2+6+5+3+2=27

27 is divisible by 9, Hence the number 5426532 is divisible by 9.

Illustration 2:- In 287286

sum of digits=2+8+7+2+8+6=33

33 is not divisible by 9, Hence the number 287286 is not divisible by 9.

Law of divisibility by number 10 :-If the last digit of given number is 0, then the given number is divisible by 10.

Illustration 1:-In 585

last digit is not 0, hence 585 is not divisible by 10.

Illustration 2:-In 56580

last digit is 0, hence 56580 is divisible by 10.

Illustration 3:-In 5856

last digit is not 0, hence 5856 is not divisible by 10. 

Law of divisibility by number 11 :-From the right side of the number (by unit digit), make groups of 3–3 digits. If there are more than 2 groups, then add them according to the even or odd place.

         If the difference of their sums is divisible by 11, then the whole number will be divisible by 11.

Illustration 1:-In 25864

From right side groups are 25 , 864

First group =864

Second group=25

difference=864-25=839

839 is not divisible by 11, Hence 25864 is not divisible by 11.

Illustration 2:-In 15937691

From right side groups are 15 , 937 , 691

First group =691+15=706

Second group=937

difference=937-706=231

231 is divisible by 11, Hence 15937691 is divisible by 11.

                                                OR

Add the digits at even and odd places of the number. If the difference of both the sum is 0 or divisible by 11, then the whole number is divisible by 11.

Illustration 1:-In 2878667

First group=2+7+6+7=22

Second group=8+8+6=22

difference=22-22=0

Hence 2878667 is divisible by 11.

(because difference is 0)

Law of divisibility by number 12 :-If given number is divisible by 3 and 4 both, then the number is divisible by 12 also.

Illustration 1:- In 6532

sum of digits=6+5+3+2=16

16 is not divisible by 3, so the number 6532 is not divisible by 3.

In 6532,  the number made up of last two digits is 32. Which is divisible by 4, Hence the number 6532 is also divisible by 4.

Hence the number 6532 is not divisible by 12.

(because 6532 is divisible by 4 but not divisible by 3).

Illustration 2:- In 642

sum of digits=6+4+2=12

12 is divisible by 3, so the number 642 is divisible by 3.

In 642,  the number made up of last two digits is 42. Which is not divisible by 4, Hence the number 642 is not divisible by 4.

Hence the number 642 is not divisible by 12.

(because 642 is divisible by 3 but not divisible by 4).

Illustration 3:- In 864

sum of digits=8+6+4=18

18 is divisible by 3, so the number 864 is divisible by 3.

In 864,  the number made up of last two digits is 64. Which is divisible by 4, Hence the number 864 is also divisible by 4.

Hence the number 864 is divisible by 12.

(because 864 is divisible by 3 and 4 both).

Law of divisibility by number 13 :-From the right side of the number (by unit digit), make groups of 3–3 digits. If there are more than 2 groups, then add them according to the even or odd place.

         If the difference of their sums is divisible by 13, then the whole number will be divisible by 13.

Illustration 1:-In 15724582

From right side groups are 15, 724 , 582

First group =582+15=597

Second group=724

difference=724-597=127

127 is not divisible by 13, Hence 15724582 is not divisible by 13.

Illustration 2:-In 44408

From right side groups are 44 , 408

First group =408

Second group=44

difference=408-44=364

364 is divisible by 13, Hence 44408 is divisible by 13.

Law of divisibility by number 14 :-If given number is divisible by 2 and 7 both, then the number is divisible by 14 also.

Illustration 1:-In 315724582

last digit is 2, so 315724582 is divisible by 2 

and

From right side groups are 315 , 724 , 582

First group =582+315=897

Second group=724

difference=897-724=173

173 is not divisible by 7, Hence 315724582 is not divisible by 7.

Hence the number 315724582 is not divisible by 14 .

(because 315724582 is divisible by 2 but not divisible by 7)

Illustration 2:-In 742385 

last digit is 5, so 742385 is not divisible by 2

and

From right side groups are 742 , 385

First group=385

Second group=742

difference=742-385=357

357 is divisible by 7, Hence 742385 is divisible by 7

Hence the number 742385 is not divisible by 14.

(because 742385 is divisible by 7 but not divisible by 2)  

Illustration 3:-In 22386

last digit is 6, so 22386 is divisible by 2 

and 

From right side groups are 22 , 386

First group =386

Second group=22

difference=386-22=364

364 is divisible by 7, Hence 22386 is divisible by 7.

Hence the number 22386 is divisible by 15 also.

(because 22386 is divisible by 2 and 7 both)

Law of divisibility by number 15 :-If given number is divisible by 3 and 5 both, then the number is divisible by 15 also.

Illustration 1:- In 186

sum of digits=1+8+6=15

15 is divisible by 3, so the number 186 is divisible by 3.

last digit is not 0 or 5, hence 186 is not divisible by 5. 

Hence the number 186 is not divisible by 15.

(because 186 is divisible by 3 but not divisible by 5)

Illustration 2:- In 585

sum of digits=5+8+5=18

18 is divisible by 3, so the number 585 is divisible by 3.

last digit is 5, hence 585 is divisible by 5.

Hence the number 585 is divisible by 15 also.

(because 585 is divisible by 3 and 5 both)

Illustration 3:- In 350

sum of digits=3+5+0=8

8 is not divisible by 3, so the number 350 is not divisible by 3.

last digit is 0, hence 350 is divisible by 5.

Hence the number 350 is not divisible by 15.

(because 350 is divisible by 5 but not divisible by 3)

Law of divisibility by number 16 :-If the number made up of unit, tens, hundred and thousand digit is divided by 8, then the whole number is divided by 8.

                                                                    or 

If the number made up of last four digits of given number is divided by 16, then the whole number is divided by 16.

Illustration 1:- In 546532,  the number made up of last four digits is 6532. Which is not  divisible by 16, Hence the number 546532 is not divisible by 16.

Illustration 2:- In 2487286,  the number made up of last four digits is 7286. Which is not divisible by 16, Hence the number 2487286 is not divisible by 16.

Illustration 3:- In 65878688,  the number made up of last four digits is 8688. Which is divisible by 16, Hence the number 65878688 is also divisible by 16.

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