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



of a and b. Evidently I must divide d e. Let this condition be satis- 

 fied and determine A and B so that Aa Bb = I. Multiply the last 

 equation by (d e)/l. Then 



d e d e 



Aa --- - -- (- d = Bb --- \- e 



L L 



is an answer. Other answers follow by adding any multiple of the l.c.m. 

 M of a, b. Next, let there be three divisors a, b, c and corresponding 

 remainders d, e, f. By the first problem find a number g having the re- 

 mainders d and e when divided by a and b, and then a number h having the 

 remainders g and/ when divided by M and c. From the answer h we obtain 

 others by adding any multiple of the l.c.m. of a, b, c. 



* A. G. Kastner," * Liidicke 100 and C. Button 101 treated problems on 

 the Julian period. To find the year x of Christ in which the solar and 

 lunar cycles are 18 and 8, and Roman indiction is 10, Hutton noted that 

 the year before the Christian era was the ninth of the solar cycle, first of 

 the lunar and third of the indiction. Hence the remainders on dividing 

 z + 9, z -f 1, # + 3 by 28, 19, 15 respectively (the periods of the solar, 

 lunar and indiction cycles) must be 18, 8, 10. Thus x = 7980p + 1717. 



To apply 102 the rule in J. Keill's Astronomy Lectures, p. 380, divide 

 18-4845 + 8-4200 + 10-6916 by 7980; the remainder 6430 is the year of 

 the Julian period; subtract 4713, the Julian year at the birth of Christ. 



A. Thacker 103 proved the last rule, starting as had Hutton. 101 



The least number 104 with the remainders 1, 2, 3, 4, 5 when divided by 

 2, 3, 4, 5, 6 is 60 - 1 [L. Pisano 82 ]. 



R. Robinson 105 found a number x which has the remainders 19, 18, 

 -, 1 when divided by 20, 19, -, 2. Since 



z = 2a + l = 36 + 2= ... = 20A + 19, b = 2m - 1, a = 3m - 1; 

 then use x = 4c + 3, etc. Hence x = 2327925605 1, the least being 

 given by B = 1. 



J. L. Lagrange 106 determined n so that it shall have given remainders 

 N, NI, Nz, " -, when divided by M, MI, M z , - respectively. Let P be 

 the l.c.m. of M, MI, M 2 , ; Q that of M, M 2 , M 3 , - (omitting MI) ; 

 Qi that of M, MI, M 3 , (omitting M 2 ) ; etc. Then seek (Lagrange 17 ) 

 integers ju, v, m, vi, such that 



99 Angewandte Math., in der Chronologie. 



100 Archiv der Math, (ed., Hindenburg), 2, 1745, 206. 



101 The Diarian Repository, or Math. Register, by a Society of Mathematicians, London, 



1774, 306; The Diarian Miscellany, extracted from Ladies' Diary, London, 2, 1775, 

 33-4; Leybourn's Math. Quest, proposed in Ladies' Diary, 1, 1817, 232-3. 



102 Ladies' Diary, 1735, 33-4, Quest. 175. 



103 A Miscellany of Math. Problems, Birmingham, 1, 1743, 167-8. 



104 Ladies' Diary, 1749, 21, Quest. 296; Diarian Repository ... by a Society of Mathe- 



maticians, London, 1774, 501-2; C. Button's Diarian Miscellany, 2, 1775,264-5; Ley- 

 bourn's Math. Quest. L. D., 2, 1817, 2. 



105 The Gentleman's Diary, or Math. Repository, 1748; A. Davis' ed., London, 1, 1814, 154-5. 

 108 M6m. Acad. Roy. Sc. Berlin, 23, anne"e 1767 (1769); Oeuvres, II, 519-20. 



