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

GENERAL PROBLEM OF REMAINDERS 243 The rationale of this method is not difficult. I shall quote it from the book of Datta and Singh: Starting with the considera- tion of the first two divisors, we have N = a₁x₁ + r₁ = a₂x₂ + r₂. By the method described before, we can find the minimum value α of x₁ satisfying this equation. Then the minimum value of N will be a₁α + r₁. Hence the general value of N will be given by N = a₁ (a₂t + α) + r₁ = a₁a₂ t + a₁α + r₁ where t is an integer. Thus a₁α + r₁ is the remainder left on dividing N by a₁ a₂ as stated by Āryabhaṭa I and Brahmagupta. Now taking into consideration the third condition, we have N = a₁a₂t + a₁αr₁ = a₃x₃ + r₃ which can be solved in the same way as before. Proceeding in this way successively, we shall ultimately arrive at a value of N satisfying all the conditions ; Pṛthūdaka Svāmī remarks : Wherever the reduction of two divisors by a common measure is possible, there 'the product of the divisors' should be understood as equivalent to the product of the divisor corresponding to the greater remainder and quotient of the divisor corresponding to the smaller remainder as reduced (i.e. divided) by the common mea- sure.¹ When one divisor is exactly divisible by the other, then the greater remainder is the (required) remainder and the divisor corresponding to the greater remainder is taken as 'the product of the divisors'. (The truth of) this may be investigated by an intelligent mathematician by taking several symbols. As an illustration we shall take up a problem quoted by Bhāskara II in his Bījagaṇita, and which in its solution follows the method of Āryabhaṭa 1. Pṛthūdaka Svāmī while commenting on serveral verses from Brahmagupta (BrSpSi. XVIII. 3-6)

1. i.e., if p be the L.C.M. of a, and a2, the general value of N satisfying the above two conditions will be N = pt + a₁a + r, instead of N = a₁a₂t + a₁a + r₁.