42 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



divisible by 73, employ 



31=0-73 + 31, 73 = 2-31 + 11, 31 = 2-11 + 9, 

 11 = 1-9 + 2, 9 = 4-2 + 1. 



The least remainder of odd rank is 1. Choose a number a = 5 such that 

 a-1 3 is divisible by the last divisor 2, the quotient being 1. By use of 

 5, 1 and the quotients 2, 2, 1, 4 after the first, we derive 



2 2 1 451 



172 = 2-73+26, 73 = 2-26+21, 26 = 1-21+5, 21 = 4-5+1. 



A smaller answer than 172 is given by 172 2-73 = 26. 



In the second example, 63z + 7 is to be made a multiple of 23. Here 



63 = 2-23 + 17, 23 = 1-17 + 6, 17 = 2-6 + 5, 

 6 = 1-5 + 1, 5 = 4-1 + 1, 



the division being carried an extra step so as to yield the last remainder of 

 odd rank. Here a = 1 makes a-1 + 7 divisible by the last divisor 1. 

 Discarding the first quotient, we have 1, 2, 1, 4, 1, 8 and then get 51, 38 

 13, 12. Since 51 = 2-23 + 5, an answer is 5. 



Bha"scara Achdrya 4 (born, 1114) gave detailed methods of finding a 

 pulverizing multiplier (Cuttaca) such that if a given dividend be multiplied 

 by it and the product added to a given additive quantity, the sum will be 

 exactly divisible by a given divisor. 



First ( 248-252), we reduce the dividend, divisor and additive by 

 their g.c.d. If a common divisor of the dividend and divisor does not 

 divide also the additive, the problem is impossible. 



Next ( 249-251), divide mutually the reduced dividend and divisor 

 until the remainder unity is obtained. Write the quotients in order, 

 after them write the additive, and after it zero. To the last term add the 

 product of the penult by the next preceding number. Reject the last term 

 and repeat the operation until only two numbers are left. The first of these 

 is abraded by the reduced dividend, and the remainder is the desired quo- 

 tient. The second of the two, abraded by the reduced divisor, is the 

 desired multiplier. 



Example ( 253) : Dividend 17, Divisor 15, Additive 5. The quotients 

 are 1, 7, so that the series is 1, 7, 5, 0. Since + 7-5 = 35, the new series 

 is 1, 35, 5. The final series is 40, 35. Abrading them by multiples of 17 

 and 15 respectively, we get 6 and 5 as the desired quotient and multiplier 

 [17-5 + 5 = 15-6]. 



4 Lilavati (Arithmetic), Ch. 12, 248-266, Colebrooke 1 , pp. 112-122. [It is nearly word 

 for word the same as Ch. II of Bhascara's Vija-ganita (Algebra), 53-74, Colebrooke, 1 

 pp. 156-169; Bija Ganita or the Algebra of the Hindus, transl. into English by E. 

 Strachey of the Persian transl. of 1634 by Ata Alia Rasheedee of Bhascara Acharya, 

 London, 1813, Ch. / of Introduction, pp. 29-36. Lilawati or a Treatise on Arith. & 

 Geom. by Bhascara Acharya, transl. from the original Sanskrit by John Taylor, Bom- 

 bay, 1816, Part III, Sect. I, p. Ill; the Persian transl. in 1587 by Fyzi omitted the 

 chapters on indeterminate problems. Lilawati was the name of Bhascara's daughter.] 



