ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

244 BRAHMAGUPTA AS AN ALGEBRAIST
observes that such problems were very popular amongst the an-
cient Indian mathematicians.
Problem : To find a number N which leaves remainders 5,
4,3,2 when divided by 6,5,4,3 respectively.
That is to solve the equations :
N = 6x + 5 = 5y + 4 = 4z + 3 = 3w + 2.
We have since N = 6x + 5 = 5y + 4,
x = (5y - 1) / 6
But x must be integral, so y = 6t + 5, x = 5t + 4
Hence N = 30t + 29
Again N = 30t + 29 = 4z + 3
Therefore, t = (2z - 13) / 15
Since t must be integral, we must have z = 15s + 14;
hence t = 2s + 1. Therefore
N = 60s + 59.
The last condition is identically satisfied. The method given
here is the one followed by Pṛthūdaka Svāmī.
Thus when N = 60s + 59 = 6x + 5
x = (60s + 54) / 6 = 10s + 9 ...(1)
Again, when N = 60s + 59 = 5y + 4,
y = (60s + 55) / 5 = 12s + 11
Again when N = 60s + 59 = 4z + 3
z = (60s + 56) / 4 = 15s + 14
Lastly, when N = 60s + 59 = 3w + 2.
w = (60s + 57) / 3 = 20s + 19,
Varga Prakṛti or Kṛti Prakṛti or Square-Nature
The word varga-prakṛti (literally meaning 'square-nature')
has been given by Indian algebraists to the indeterminate quadra-
tic equation
Nx² ± c = y²