42 JIISTOBY OF THE THEORY OF JNUMBEES. [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-I — 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 21 4 51

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, 63# + 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-I + 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 Acharya4 (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 quotient. 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 0 + 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].

<Lflavati (Arithmetic), Ch. 12, §§248-266, Colebrooke1, 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. 2&-S6. Lilawati or a Treatise on Arith. & Geom. by Bhascara Acharya, transl. from the original Sanskrit by John Taylor, Bombay, 1816, Part Ill^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.]