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