## Kevalaih Saptakam Gunyat

### Sanskrit Name:

केवलैः सप्तकम् गुण्यात्

### English Translation:

For 7 the Multiplicand is 143 (Kevala: 143, Sapta: 7,).

### Usage:

On the basis of 1/7, without any multiplication we can calculate 2/7, 3/7, 4/7, 5/7 and 6/7. For that 1/7=0.142857 is to be remembered. But since remembering 0. 142857 is difficult we remember Kevala(143). This is only use of this sutra (for remembrance).

1/7 = 0.142857

and By Ekanyuna corollary, 143 x 999 = 142857

We need to remember 142857 for following reasons:

- 2/7 = 0.285714
- 3/7 = 0.428571
- 4/7 = 0.571428
- 5/7 = 0.714285
- 6/7 = 0.857142

All the values are in cyclic order.

So if we can remember 1/7 then we can obtain 2/7, 3/7 and etc as

- 2/7: last digit of answer must be 4 (2*7=14)
- 3/7: last digit of answer must be 1 (3*7=21)
- 4/7: last digit of answer must be 8 (4*7=28)
- 5/7: last digit of answer must be 5 (5*7=35)
- 6/7: last digit of answer must be 2 (6*7=42)

Also Read => More Division Sutras in Vedic Mathematics

# Vestanas (Osculators)

### Sanskrit Name:

वेष्तनs

### English Translation:

Osculators.

### Usage:

Vestanas Sutra of Vedic Mathematics is used whenever we need to find whether given number is exactly divisible by divisor.

### Process:

It uses the concept of Ekadhika Purvena.

Multiply the last digit of dividend by Ekadhika and add previous/remaining digits to that product. Continue this till a comparatively smaller number is obtained which can be seen easily as exactly divisible or not divisible by divisor.

### Small Recap:

1/7 = 7/49; Ekadhikas for 7 is 5

1/9; Ekadhikas for 9 is 1

1/19; Ekadhikas for 19 is 2

1/13 = 3/39; Ekadhikas for 13 is 4 and so on.

Above Ekadhikas are called as **Positive Osculators**. *Later we will see Negative Osculators.*

### Examples:

Lets check whether 21 is divisible by 7.

For 7, Ekadhika(positive osculator) is 5

So as per the mentioned process, multiply 5 with 1 and add 2 to the product.

- 21; 1×5+2 = 7 (Divisible by 7)
- 91; 1×5+9 = 14 (Divisible by 7). Can be continued further as

14; 4×5 + 1 = 21; and

21;1×5+2 = 7 - 112; 2×5+11= 21. (seen earlier)
- 2107; 7×5 + 210 = 245

245; 5×5+24= 49 (Divisible by 7 or continue further).

For 13, Ekadhika(positive osculator) is 4

- 2197; 7×4+ 219 = 247

247; 7×4+24= 52

52; 2×4+5=13 (Divisible by 13)

For 29, Ekadhika (positive osculator) is 3

- 2434521; 1×3+243452 = 243455

243455; 5×3+24345 = 24360

24360; 0x3+2436 = 2436

2436; 6×3 + 243 = 261

261; 1×3+ 26 = 29 (Divisible by 29)

For 23; Ekadhikas (positive osculator)is 7

- 529; 9×7+52= 115

115; 5×7+11=46

46; 6×7+4 = 46 (Cannot go further to 23 but divisible by 23).

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