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

INDETERMINATE EQUATIONS 225 quotient or phala. In later years, Mahāvīra has called the unknown number (x) as rāśi. Preliminary Operations in Kuṭṭaka-Karma Usually it has been suggested that in order that an equation of the form by−ax= ± c or by+ax= ± c may be amenable to solution, the two numbers a and b must not have a common divisor; for otherwise, the equation would be absurd, unless the number c had the same common divisor. So before the rules which we shall give hereafter, could be applied, the numbers a, b, c must be made prime (dṛḍha or firm; niccheda or having no divisor, or nirapavarta, meaning irreducible to each other. In this connection Bhāskara I writes : The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.¹ Similarly we find in the writings of Brahmagupta : Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other.² Āryabhaṭa's Rule : Āryabhaṭa I is probably the first Indian writer on this subject, but the operation given by him is rather obscure. His disciple Bhāskara I has given the solution of inde- terminate equations of the first degree in more satisfactory langu- age. We shall give here the translation of Āryabhaṭa's verse from the Āryabhaṭīya, as rendered by Bibhutibhusan Datta, because other translations of this verse do very often confuse the sense : Divide the divisor corresponding to the greater remain- der by the divisor corresponding to the smaller remain-

1. भूदिनेष्टगणा-न्योन्य भक्तशेषेण भाजितौ । हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोर्विदुः —MBh, I. 41
2. हत्योः परस्परं यच्छेषं गुणकारभागहारकयोः । तेन हृतौ निश्छेदौ तावेव परस्परं हत्योः । —BrSpSi. XVIII. 9.