CHAP. II] CHINESE PROBLEM OF REMAINDERS. 57 



J. G. Zehfuss 68 gave the formula of Cauchy 60 and noted that, if 

 H = a^n . . . j and if A is not divisible by the prime a, B not by /?, , 

 then 



For A = B = = a, let the left member become k. Then ax = b 

 (mod /i) has the root kb/a. It also has the root (1 AB - - -)b/a, where 



/ ( a IVY" 



A = ( 1 + a^- ^- ) = (mod a"), 

 \ tt a / 



IV \ n 



where a a is the least positive residue of a modulo a, since, by Wilson's 

 theorem, a a + (a 1) ! a is divisible by the prime a. 



M. F. Daniels 69 noted that, if PI---P B = 1 (mod &) by Wilson's 

 generalized theorem, then piX = 1 (mod k) has the root pi---p;_i 

 PH-I- -Pn- Further, if k = p"q^- and if aci = 1 (mod p), ac 2 = 1 (mod 

 q), - - -, then ax = 1 (mod A;) has the root 



x = - {1 - (1 - aci)'(l - aczY- '}- 

 a 



J. Perott 70 noted that if a and u are relatively prime and if a belongs to 

 the exponent t modulo u, ax = 1 (mod u) has the unique solution x = a'" 1 

 (mod u) . He admitted he was anticipated by Cauchy. 



CHINESE PROBLEM OF REMAINDERS. 



Sun-Tsu, 71 in a Chinese work Suan-ching (arithmetic), about the first 

 century A.D., gave in the form of an obscure verse a rule called t'ai-yen 

 (great generalisation) to determine a number having the remainders 2, 

 3, 2, when divided by 3, 5, 7, respectively. He determined the auxiliary 

 numbers 70, 21, 15, multiples of 5-7, 3-7, 3-5 and having the remainder 1 

 when divided by 3, 5, 7, respectively. The sum 2 70 -f 3 21 + 2 15 = 233 

 is one answer. Casting out a multiple of 3 5 7 we obtain the least answer 

 23. The rule became known in Europe through an article, "Jottings on 

 the science of Chinese arithmetic," by Alexander Wylie, 72 a part of which 

 was translated into German by K. L. Biernatzki. 73 A faulty rendition by 



68 Diss. (Heidelberg), Darmstadt, 1857; Archiv Math. Phys., 32, 1859, 422. 



69 Lineaire Congruenties, Diss., Amsterdam, 1890, 114, 90. 



70 Bull, des Sc. Math., (2), 17, I, 1893, 73^1. 



71 Y. Mikami, Abh. Geschichte Math. Wiss., 30, 1912, 32. 



"North China Herald, 1852; Shanghai Almanac for 1853. Cf. remark by G. Vacca, 

 Bibliotheca Math., (3), 2, 1901, 143; H. Cordier, Jour. Asiatic Soc., (2), 19, 1887, 358. 



73 Jour, fur Math., 52, 1856, 59-94. French transl. by O. Terquem, Nouv. Ann. Math., (2), 

 1, 1862 (Bull. Bibl. Hist.), 35-44; 2, 1863, 529-540; and by J. Bertrand, Journal des 

 Savants, 1869. Cf. Matthiessen. 79 



