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उपसूत्र २. शिष्यते शेषसंञः

(Upasūtra 2. śiṣyate śeṣasṃjñaḥ) - The remainder remains constant.

The Upasūtra: śiṣyate śeṣasṃjñaḥ (The remainder remains constant) is used for fractions: Division by a composite number, the composition of which are in ratio with each other. It observes that if the divisors are in ratio, the remainder of one of the divisors will be the same as the remainder for the other divisors.
 
As an illustration, let us use this Upasūtra for:
103 ÷ 64
  Steps 103 ÷ 64
1. Find the Ratio of the composition of the Divisor. In this case, 64 = 2 × 4 × 8
where, 2:4 = 4:8 = 8:64 = 1:2
2. Find the Quotient and Remainder by one of the smaller proportions. In this case,
Let us consider 4, then 103/4 =
   3
25 -
   4
Quotient = 25
Remainder = 3
3. The Remainder being the same, find the desired Quotient by dividing the quotient with the proportional Ratio. In this case,
So, the desired Quotient = 25/16 = 1
And, Remainder = 9
4. In case of additional remainders, re-evaluate the final remainder by adding both of them. In this case, the desired Remainder =
 3    9   3 + 36
-- + -- = ------
64   16     64
 
That gives us:
  39
1 --
  64, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
1063 ÷ 27
  27 = 3 × 9, where 3:9 = 9:27 = 1:3
Then, 1063/9 = 118, Remainder = 1
 
Again, 118/3 = 39, Remainder = 1
 
Thus, 1063 ÷ 27 =
     1    1       10
39 (-- + --) = 39 --
     3   27       27
 
This Upasūtra is extremely useful for Divisors of the form x3. Again, for something like:
70804 ÷ 343
  343 = 7 × 49, where 7:49 = 49:343 = 1:7
Then, 70804/7 = 10114, Remainder = 6
 
Again, 10114/49 = 206, Remainder = 20
 
Thus, 70804 ÷ 343 =
      6    20        146
206 (--- + --) = 206 ---
     343   49        343
But, why does it work? For this Upasūtra (śiṣyate śeṣasṃjñaḥ), let us consider the following:
  For any successive numbers, with ratio k:
a:b:c:d
Then, b = ka, c = k2a, d = k3a
 
When any number, x is divided by these successive numbers:
x/b = x/(ka)
x/c = x/(k2a)
x/d = x/(k3a)
And so on.
 
It should be noted that not only does this series have a constant Remainder, the Quotients are also in (reducing) proportions in the same ratio.
 
For example,
If x/a gives quotient Q, and remainder R
x     R
- = Q -
a     a
     x    x   x   1   1    R
And, - = -- = - . - = - (Q -)
     b   ka   a   k   k    a
     x    x    x    1    1    R
And, - = --- = - . -- = -- (Q -)
     c   k2a   a   k2   k2    a
     x    x    x    1    1    R
And, - = --- = - . -- = -- (Q -)
     d   k3a   a   k3   k3    a
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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