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