344 History of the Theory of Numbers. [Chap, xii 
E. NanneF employed ri=Oi — aoX, r2 = a2—riX,. . . (a-<10). Then, if 
r„ = 0, A'' = 10"a„4- . . . +10ai+Oo is divisible by lOx-fl and the quotient has 
the digits r„_i, r„_2, . . ■ , 7*i, Qo- The cases x = 1, 2 are discussed and several 
tests for 7 deduced. For a:= 1/3, we conclude that, if r„ = 0, N is divisible 
by 13 and the digits of the quotient are r„_i/3, . . . , r^/S, ao/3. 
A. Chiari^ employed D'Alembert's^ method for 10+6, 6 = 3, 7, 9. 
G. Bruzzone^^ 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 = 10'', form the 
groups of r digits counting from the right of N, and multiply the successive 
groups (from the right) hy l,y,y^,. . . 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 = lO^, we must change alternate signs before adding. For practical use, 
take y = l. 
Fr. Schuh^^ gave three methods to determine the residue of large numbers 
for a given modulus. 
Stuyvaert^^ let a, 6, ... be the successive sets of n digits of A'' to the baseB, 
so that iV = a+6jB'*+c52''+ rj.^^^ ^ -^ (^^.^isibie ^^y ^ factor D of B''=pR'' 
if and only if a=^bR''+cR~''^ ... is divisible by D. For R = l, B = 10, 
n = 1, 2, . . ., we obtain tests for divisors of 9, 99, 11, 101, etc. A divisor, 
prime to B, of niB+l divides N = a+bB if and only if it divides h—ma. 
Further Papers Giving Tests for a Given Divisor d. 
J. R. Young and Mason for d = l, 13 [Pascal^], Ladies' Diary, 1831, 34-5, Quest. 
1512. 
P. Gorini [Pascal^], Annali di Fis., Chim. Mat., (ed., Majocchi), 1,1841, 237. 
A. Pinaud for d = l, 13, Mem. Acad. Sc. Toulouse, 1, 1844, 341, 347. 
*Dietz and Vincenot, Mem. Acad. Metz, 33, 1851-2, 37. 
Anonymous writer for d = 9, 11, Jour, fiir Math., 50, 1855, 187-8. 
*H. Wronski, Principes de la phil. des math. Cf. de Montferrier, Encyclop^die 
math., 2, 1856, p. 95. 
O. Terquem for d^l9, 23, 37, 101, Nouv. Ann. Math., 14, 1855, 118-120. 
A. P. Reyer for d = l, Archiv Math. Phys., 25, 1855, 176-196. 
C. F. Lindman for d = l, 13, ibid., 26, 1856, 467-470. 
P. Buttel for d = 7, 9, 11, 17, 19, ibid., 241-266. 
De Lapparent [Herter^^], Mem. soc. imp. sc. nat. Cherbourg, 4, 1856, 235-258. 
Karwowski [Pascal^], 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 = l-n, Proc. Roy. Soc. London, 10, 1859-60, 208. 
A. Niegemann for d = 10'"-n+a, 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. 
»/6r(f., 14, 1907-8, 35-7. 
"/6td., 15, 1908-9, 119-123. 
"Supplem. De Vriend der Wiskunde, 24, 1912, 89-103. 
"Les Nombres Positifs, Gand, 1912, 59-62. 
