344 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XII
E. Nannei53 employed r1=o.1 — a0x, r2 = a2—r1x,. . . (:c<10). Then, if rn=0, N = lQnan+ . • .+10a!+a0is divisible by lOz+1 and the quotient has the digits rn_l3 rn_2, . . . , r1; a0. The cases x = 1, 2 are discussed and several tests for 7 deduced. For a; = 1/3, we conclude that, if rn=0, N is divisible by 13 and the digits of the quotient are rn_!/3, . . . , ^/S, a0/3.
A. Chiari54 employed D'Alembert's5 method for 10+&, 6 = 3, 7, 9.
G. Bruzzone55 noted that, to find the remainder R when N is divided by an integer x of r digits, we may choose y such that x+y^W, form the groups of r digits counting from the right of N, and multiply the successive groups (from the right) by 1, y, y2, . . . or by their residues modulo x] then R equals the remainder on dividing the sum of the products by x. If we choose x — y = 10r, we must change alternate signs before adding. For practical use,
Fr. Schuh56 gave three methods to determine the residue of large numbers for a given modulus.
Stuyvaert57 let a, 6, ... be the successive sets of n digits of TV to the base 5, so that N = a+bBn+cB2n+ .... Then N is divisible by a factor D of Bn =?Rn if and only if a*=bRn+cR2n± ... is divisible by D. For E = l, £ = 10, n = l, 2, . . ., we obtain tests for divisors of 9, 99, 11, 101, etc. A divisor, prime to B, of mB+l divides N — a-\-bB if and only if it divides b — ma.
FURTHER PAPERS GIVING TESTS FOR A GIVEN DIVISOR d.
J. R. Young and Mason for d = 7, 13 [Pascal4], Ladies' Diary, 1831, 34-5, Quest.
1512.
P. Gorini [Pascal4], Annali di Fis., Chim. Mat., (ed., Majocchi), 1,1841, 237. A. Pinaud for d = 7, 13, M&n. Acad. Sc. Toulouse, 1, 1844, 341, 347. *Dietz and Vincenot, M&n. Acad. Metz, 33, 1851-2, 37. Anonymous writer for d = 9, 11, Jour, fur Math., 50, 1855, 187-8. *H. Wronski, Principes de la phil. des math. Cf. de Montferrier, Encyclopedic
math., 2, 1856, p. 95.
0. Terquem for d£l9, 23, 37, 101, Nouv. Ann. Math., 14, 1855, 118-120. A. P. Reyer for d = 7, Archiv Math. Phys., 25, 1855, 176-196. C. F. Lindman for d = 7, 13, ibid., 26, 1856, 467-470. P. Buttel for d = 7, 9, 11, 17, 19, ibid., 241-266.
De Lapparent [Herter14], Mem. soc. unp. sc. nat. Cherbourg, 4, 1856, 235-258. Karwowski [Pascal4], Ueber die Theilbarkeit . . ., II, Progr., Lissa, 1856. *D. van Langeraad, Kenmerken van deelbarheid der geheele getallen, Schoonho-
ven, 1857.
Flohr, Ueber Theilbarkeit und Reste der Zahlen, Progr., Berlin, 1858. V. Bouniakowsky for d = 37, 989, Nouv. Ann. Math., 18, 1859, 168. Elefanti for d = 7-13, Proc. Roy. Soc. London, 10, 1859-60, 208. A. Niegemann for d = 10m-n-fa, Archiv Math. Phys., 38, 1862, 384-8. J. A. Grunert for d = 7, 11, 13, ibid., 42, 1864, 478-482. V. A. Lebesgue, Tables diverses pour la decomposition des nombres, Paris, 1864,
p. 13. _
"II Pitagora, Palermo, 13, 1906-7, 54-9.
»Ibid., 14, 1907-8, 35-7.
"/&«*., 15, 1908-9, 119-123.
"Supplem. De Vriend der Wiskunde, 24, 1912, 89-103.