Addition

VEDIC MATHEMATICS - 3

Square Root of Exact Square

Introduction

To find out the square root of exact squares first of all we should know if the

number is a perfect square or not.

Introduction

To find out if any given number is an exact square we should follow the following

fundamental rules:

A perfect square ends in 0,1,4,5,6 and 9.

A number is a not perfect square if it ends in 2, 3, 7 or 8

The number should end in even number of zeros.

If the number ends in 6, then its second last digit should be odd.

If the number does not end in 6 then its second last digit should be even.

If the number is even its last two digits should be divisible by 4.

If the given number has n digits then its square root will have, n/2 digits if n is even, or (n+1)/2

digits if n is odd.

Let us study the squares of the first 10 natural numbers

Now let us study the squares of the first 10 natural numbers.

Number Square Last digit of square

1 1 1

2 4 4

3 9 9

4 16 6

5 25 5

6 36 6

7 49 9

8 64 4

9 81 1

10 100 00

From the above table we conclude that:

If the square ends in 1 then the last digit of its square root will either be 1 or 9.





Divisibility by 3 - Examples

1. 12345 = 1 + 2 + 3 + 4 + 5 = 15 and 15 is divisible by 3, the number 12345 is

divisible by 3.

2. 345678 = 3 + 4 + 5 + 6 + 7 + 8 = 33, and 33 is divisible by 3, the number 345678 is

divisible by 3.

3. 6547891 = 6 + 5 + 4 + 7 + 8 + 9 +1 = 40, and 40 is not divisible by 3, the number is

not divisible by 3.

4. 85856858 = 8 + 5 + 8 + 5 + 6 + 8 + 5 + 8 = 53 = 5 + 3 = 8 and 8 is not divisible by 3,

the number is not divisible by 3.

Divisibility by 4

If the number formed by the last two digits is evenly divisible by 4, the entire

number is evenly divisible by 4.

Add the digits of the given number and if the sum is divisible by 3, then the number

is divisible by 3.

Divisibility by 4- Examples

1. 76305144 the last two digits is 44, and 44 is divisible by 4 , the entire number is

divisible by 4.

2. 123456 the last two digits is 56, and 56 is divisible by 4, the entire number is

divisible by 4.

3. 4567898 the last two digits is 98, and 98 is not divisible by 4, the entire number is

not divisible by 4.

4. 568774 the last two digits is 74, and 74 is not divisible by 4, the entire number is not

divisible by 4.

Divisibility by 5

If the number ends in 0 or 5. It is evenly divisible by 5.


1. 76305 the last digit is 5, so the entire number is divisible by 5.




Divisibility by 6

If the sum of the digits of an even number is 3 , 6 or 9, the entire number is divisible

by 6.


1. 12744 = 1 + 2 + 7 + 4 + 4 = 18 = 1 + 8 = 9, the number is divisible by 6.

2. 1234560 = 1 + 2 + 3 + 4 + 5 + 6 + 0 =21 = 2 + 1 = 3 the number is divisible by 6.

3. 4567898 = 4 + 5 + 6 + 7 + 8 + 9 + 8 = 43 = 4 + 3 = 7 the number is not divisible by 6.

4. 56877, it is a odd number so the number is not divisible by 6.

Divisibility by 7

Mark off groups of three digits, starting from the right. Add the alternate groups.

Find the difference between the two sums thus obtained. If the difference is zero or

multiple of 7, the given number is divisible by 7.

Divisibility by 7 – Example- 1

58556344

58 556 344

Add the first and third group, i.e., 58 and 344 = 402

Find the difference of 402 and 556 = 154

Check whether 154 is a multiple of 7.

7 x 22 = 154, so the entire number is divisible by 7.


2352

2 352


Check whether 350 is a multiple of 7.

7 x 50 = 350, so the entire number is divisible by 7.


2897858961

2 897 858 961

Add the first and third group, i.e., 2 and 858 = 860

And second and fourth group i.e., 897 and 961 = 1858


Check whether 998 is a multiple of 7or not.

So the entire number is not divisible by 7.

Divisibility by 8

If the number formed by the last three digits of the given number is evenly divisible

by 8, the entire number is evenly divisible by 8. Of course the first requirement is

that the given number is even. A further test can be made by inspection is to see

whether or not the given number is evenly divisible by 4. If it is not, then it cannot

be evenly divided by 8.

Divisibility by 8 – Examples

1. 12744 = The last two digits of the number is 44, 44 is divisible by 4, and now check

for 744 744/8, yes it is divisible by 8.

2. 123456 = The last two digits of the number is 56, 56 is divisible by 4, and now check

for 456 456/8, yes it is divisible by 8.

3. 456789 = It is an odd number, so it is not divisible by 8.

4. 56878 = The last two digits of the number is 78, 78 is not divisible by 4, and the

entire number is not divisible by 8.

Divisibility by 9

Any number is divisible by 9, if the sum of its digits is divisible by 9.


1. 69867 = 6 + 9 + 8 + 6 + 7 = 36 = 9 , so the entire number is divisible by 9.

2. 3188646 = 3 + 1 + 8 + 8 + 6 + 4 + 6 = 36 = 9 , so the number is divisible by 9.

3. 123456 = 1 + 2 + 3 + 4 + 5 + 6 =21 = 3 , so the number is not divisible by 9.

4. 50541 = 5 + 0 + 5 + 4 + 1 = 15 = 6 , so the number is not divisible by 9.

Divisibility by 10

All the numbers that end in 0 are divisible by 10.


698670 , the number ends with 0, so it is divisible by 10.

314560 , the number ends with 0, so it is divisible by 10.

987988 , the number ends with other than 0, so it is not divisible by 10.

156487 , the number ends with other than 0, so it is not divisible by 10.

Divisibility by 11

If the difference of the sum of the evenly placed digit from the sum of the oddly

placed digits or vice versa gives zero, 11 or multiple of 11, then the number is

divisible by 11.

Divisibility by 11 – Example - 1

470448

Evenly placed digits 7 + 4 + 8 = 19

Oddly placed digits 4 + 0 + 4 = 8

Difference = 19 – 8 = 11, which is divisible by 11, so the number is divisible by 11.


940896

Evenly placed digits 4 + 8 + 6 = 18

Oddly placed digits 9 + 0 + 9 = 18

Difference = 18 – 18 = 0, so the number is divisible by 11.


12345678

Evenly placed digits 2 + 4 + 6 + 8 = 20

Oddly placed digits 1 + 3 + 5 + 7 = 16

Difference = 20 – 16 = 4, which is not divisible by 11, so the number is not divisible

by 11.

Divisibility by Prime Numbers

To check if the given number is divisible by any prime number we use its ekadhika.

We multiply the last digit of the number by its ekadhika and add it to the other digits

prior to the last digit. We keep doing it till we reach to a number that we alaredy

know is divisible by the given number.

Ekadhika of any prime number can be calculated by multiplying it by such a number

that gives 9 in the units place, and then we take one more than the previous digit(s)

to 9 of the answer, as the ekadhika of the number.

Ekadhika of 7 = 7 x 7 = 49, 4 (previous digit) + 1 = 5

Ekadhika of 13 = 13 x 3 = 39, 3 (previous digit ) 3 + 1 = 4

Ekadhika of 19 = 19 x 1 = 19, 1(previous digit) 1 + 1 = 2

Ekadhika of 29 = 29 x 1 = 29, 2 (previous digit) 2 + 1 = 3

Divisibility by 7

We use the ekadhika to find the divisibility by 7.

The ekadhika of 7 is 5.


prior to the last digit. We keep doing it till we reach a number that we already know

is divisible by 7.

Divisibility by 7 Example - 1

864976

Ekadhika of 7 is 5

Last digit is 6, so 6 x 5 = 30, add this to 86497, 86497 + 30 = 86527

Now last digit is 7, 7 x 5 = 35, add this to 8652, 8652 + 35 = 8687




35 is divisible by 7

So, the number 864976 is divisible by 7.


25658

Ekadhika of 7 is 5




53 is not divisible by 7

So, the number 25658 is not divisible by 7.

Divisibility by 13

We use the ekadhika to find the divisibility by 13.

The ekadhika of 13 is 4.


prior to the last digit. We keep doing it till we reach a number that we already know

is divisible by 13.


114244

Ekadhika of 13 is 4




130 is divisible by 13

So, the number 114244 is divisible by 13.


98568

Ekadhika of 13 is 4





18 is not divisible by 13

So, the number 98568 is not divisible by 13.

Addition

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