CHAPTER II. 



LINEAR DIOPHANTINE EQUATIONS AND CONGRUENCES. 



SOLUTION OF ax + by = c. 



The Hindu Aryabhatta 1 (fifth century or earlier) knew a general 

 method of solving indeterminate equations of the first degree. The original 

 of his treatise (on astronomy mainly) has been lost. Such a method of 

 solution is given in outline by Brahmegupta without the clear details of the 

 later presentation by Bhascara. 



Brahmegupta 2 (born 598 A.D.) gave the following rule to find a constant 

 " pulverizer." From the given multiplier and divisor, remove their greatest 

 common divisor (found by mutual division). The thus reduced multiplier 

 and divisor are mutually divided until the residue unity is obtained, and 

 the quotients are written in order. Multiply the residue unity by a number 

 chosen so that the product less one (or plus one, if there be an odd number 

 of quotients) shall be exactly divisible by the divisor which produced the 

 residue unity. After the above listed quotients place this chosen number 

 and after it the quotient just obtained. To the ultimate add the product 

 of the penultimate by the next preceding term [etc.]. The number found, 

 or its residue after division by the reduced divisor, is the constant pulverizer. 



Thus if 3 and 1096 are the reduced multiplier and divisor, the single 

 quotient is 365. Multiply the residue unity by the chosen number 2 and 

 add 1. Dividing the sum by 3, we get the quotient 1. Hence the series 

 is 365, 2, 1, so that the pulverizer is 1 + 2-365 = 731. [We have 

 3-731 - 1 = 2-1096.] 



Again ( 27, p. 336), let the reduced dividend [multiplier] and divisor 

 be 137 and 60, while the augment or additive quantity is 10. By reciprocal 

 division of 137 and 60, we get the quotients 0, 2, 3, 1, 1 and last two re- 

 mainders 8 and 1. Since the augment is now positive and the number of 

 quotients is odd and since 1 9 1 is divisible by 8, we select 9 as the chosen 

 number. The constant pulverizer is said to be found as before. Its product 

 by 10 is divided by 60 to give the desired multiplier 10; 10 137 -f 10 = 60-23. 



There occur various problems ( 52-60, pp. 348-360) on astronomical 

 time leading to a linear equation in two or more variables, special values 

 being arbitrarily assigned to all but two of the variables. One equation is 

 Qy 136c = 266; without detail, the constant pulverizer is said to be 2 

 and the multiplier 4 = c, whence the quotient gives y = 135. 



Mahaviracarya 3 (about 850 A.D.) gave a process essentially that due 

 to Brahmegupta, though not requiring that the initial division be continued 

 until the remainder unity is reached. To find x such that 3 la; 3 is 



1 Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhdscara, 



translated by H. T. Colebrooke, London, 1817, p. x. 



2 Brahme-sphut'a-sidd'hanta, Ch. 18 (Cuttaca = algebra), 11-14. Colebrooke, 1 pp. 330-1. 



3 Ganita-Sara-Sangraha; described by P. V. S. Aiyar, Jour. Indian Math. Club, 2, 1910, 216-8. 



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