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

242 BRAHMAGUPTA AS AN ALGEBRAIST We shall take up one of the problems posed by Brahmagupta concerning astronomy and leading to the equation :¹ 197x - 1644 y - z = 6302. Hence 1644 y + z + 6302 x = ------------------- 197 The commentator assumes z = 131. Then 1644 y + 6433 x = --------------- ; 197 hence by the usual method of the pulveriser x = 41; y = 1. General Problem of Remainders A certain type of simultaneous indeterminate equations of the first degree arise out of the general problem of remainders which may thus be stated : To find a number N which being severally divided by a₁, a₂, a₃.......aₙ , leaves as remainders r₁, r₂, r₃.........rₙ respectively. While dealing with such a case, we shall have the following series of equations : N = a₁x₁ + r₁ = a₂x₂ + r₂ = a₃x₃ + r₃ = ...... = aₙ xₙ + r . We have reasons to believe that the method of solution of these equations was known to Āryabhaṭa I. In the translation of the verse in the Āryabhaṭīya, II. 32-33 (the translation of which we have already given), the term dvicchedāgram should be translated as "the result will be the remainder corresponding to the product of the two divisors", instead of "the result will be the number corresponding to the two divisors." (the last line of the translation). This explanation is in fact given by Bhāskara I, the direct disciple and earliest commentator of Āryabhaṭa I. Such a rule is clearly stated by Brahmagupta².

1: अंशकशेषेण युतात् लिप्ताशेषात्तदन्तरादथवा । भानोर्भ'दिने द्युगणं यः कथयति कुट्टकज्ञः सः ॥ —BrSpSi. XVIII. 55 2: स्वोच्छेऽन्ययुतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् । अधिकाग्रच्छेदहतमधिकाग्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII. 5