230 The equations (2n+1) and (I. 2n+1) will then be absent and the equations (I. 2n-1) and (I. 2n) will be reduced respectively to BRAHMAGUPTA AS AN ALGEBRAIST and xn-c Giving an arbitrary integral value (t') to yn we get an in- tegral value of xn-1. Then proceeding backwards as before we calculate the values of x and y. Case (ii) Next suppose that the mutual division is stopped after having obtained an even or odd number of quotients. or Subcase (ii.): If the number of quotients obtained be even the reduced form of the original equation is ray +1=ran +1 xn+c XD-1 q2ny C ơn trị ngu tan ta Giving a suitable integral value (t) to xn as will make an integral number, ren+it+c ynti Isn or we get an integral value for yn by (2n+1). The values of x and y can then be calculated by proceeding as before. Subcase (ii.2): If the number of quotients be odd the reduc- ed form of the quotient is ran-1 Xn=7an In C Xn=an Yn-C an 1 Putting ynt, where t' is an integer, such that Yen t'-c Tan-1 a whole number, we get an integral value of x-1 by (2n). Whence can be calculated the values of x and y in integers. X₂= If xa and y-ß be the least integral solution of ax+c= by, we shall have aa+c=bB Therefore albm+a)+c=b (am +B), m being any integer. Therefore, in general, x=bm+a But we have calculated before that
