64 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



common denominator and taking a linear combination x k of the new 

 numerators such that the coefficient of x is unity. 



L. Matthiessen 111 discussed the Chinese rules in modern form. 



M. F. Daniels 69 noted that if a, b, - are relatively prime integers, 

 x = A (mod a), x = B (mod 6), have the solution 



x = (-} A + (-} B+ - (mod k = 



T. J. Stieltjes 112 noted that the congruences x = a (mod A), , 

 x = \ (mod L) have a common solution if and only if a. /3, a 7, /3 -y, 

 are divisible by (A, B), (A, C), (B, C), -, respectively, where (A, B) 

 denotes the g.c.d. of A, B. The case in which A, - , L are not relatively 

 prime in pairs can be reduced [Yih-hing 79 ]] to the contrary case by writing 

 the l.c.m. of the moduli in the form M = A'B' - -L', where A', - - , L' are 

 relatively prime in pairs and divide A, , L respectively. Then any 

 solution of the initial congruences satisfies also x = a (mod A'), , 

 x = X (mod Z/), whence x = a (mod M). Conversely, the last x satisfies 

 the initial congruences if they are solvable. 



H. J. Woodall 113 found numbers with given remainders when divided 

 by 3, 5, 7, 11, 13. 



J. Cullen 114 gave a graphical method to solve x = a (mod P), -, 

 x = X (mod L), useful when P, , L are very large. 



G. Arnoux 115 gave implicitly the theorem that, if mi, , m n are rela- 

 tively prime in pairs, M = m\- -m n , /* = Jlf/wi,-, and if ai, -, a m are 

 integers such that a*/*,- = r (mod m^ for i = 1, , w, then a^i + 

 + a n ij. n = r (mod M). Proofs were given by C. A. Laisant 116 and T. 

 Hayashi. 116 



ARTICLES ON THE PROBLEM OF REMAINDERS WITHOUT NOVELTY. 



G. S. Kliigel, Math. Worterbuch, 3, 1808, 792-800. 



J. C. Schafer, Die Wunder der Rechenkunst, Weimar, 1831, 1842, Prob. 60. 



H. Kaiser, Archiv Math. Phys., 25, 1855, 76. 



G. Dostor, ibid., 63, 1879, 224. 



V. A. Lebesgue, Exercices d'analyse nume'rique, Paris, 1859, 54-8. 



Szenic, Von der Kongruenz der Zahlen, Progr. Schrimm, 1873. 



A. Domingues, Les Mondes (Revue Hebdom. des Sciences et Arts), Paris, 55, 1881, 62. 



G. de Rocquigny, ibid., 54, 1881, 304. 



D. Marchand, ibid., 54, 1881, 437. 



NUMBER co OF POSITIVE INTEGRAL SOLUTIONS OF ax + by = n, WHERE 

 a AND 6 ARE POSITIVE AND RELATIVELY PRIME. 



P. Paoli 117 noted that if ax + by = n has integral solutions, any common 

 factor of a and b must divide n and hence can be removed from every term. 



lu Zeitschr. Math. Naturw. Unterricht, 10, 1879, 106-110; 13, 1882, 187-190. 



112 Annales Fac. Sc. Toulouse, 4, 1890, final paper, pp. 31-32. 



113 Math. Quest. Educ. Times, 73, 1900, 67. 



114 Proc. London Math. Soc., 34, 1901-2, 323-34; (2), 2, 1905, 138-141. 

 116 Arith. Graphique, Paris, 1906, 29-31. 



116 L'enseignement math., 10, 1908, 220-5; 12, 1910, 141-2. 



117 Opuscula analytica, Liburni, 1780, 114. In one place in the text and in his example,*he 



erroneously took /3 between 5/2 and 6/2, instead of positive. 



