340 History of the Theory of Numbers. [Chap, xii 
not multiples of p. Application is made to the primes p^37. Again, if 
p is a prime and 
aB-+cB+d = ak"-\-ck'-\-dk = Ak"-{-Ck'-\-Dk = mp, 
where k, k', k" are prime to p, then AB~-\-CB-[-D is divisible by p provided 
k'^ — kk" is a multiple of p. 
C. F. IMoller and C. Holten-^ would test the divisibility of n by a given 
prime p by seeking a such that ap= =*= 1 (mod 10) and subtracting from n 
such a multiple of ap that the difference ends with zero. 
L. L. Hommel"-^ made remarks on the preceding method. 
V. SchlegeP^ noted that if the di\isor to be tested ends with 1, 3, 7 or 9, 
its product by 1, 7, 3 or 9 is of the form (i = lOX+1. Then a, with the final 
digit u, is divisible by d\i ai = {a — ud)/\0 is. Then treat Oj aswe did a, etc. 
P. Otto"^ would test Z for a given prime factor p by seeking a number n 
such that if the product by n of the number formed by the last s digits of Z 
be subtracted from the number represented by the remaining digits, the 
remainder is di\'isible by p if and only if Z is. ^Material is tabulated for the 
application of the method when p<100. 
N. V. Bougaief-^" noted that a^. . .Ci to base B is divisible by D if 
fli . . .a„ to base d is divisible by D, where dB= 1 (mod D). For jB = 10 and 
Z) = 10/1+9, 1, 3, 7, we may take c? = n + l, 9?i+l, 3nH-l, 7n+5, respec- 
tively. Again, kB--\-aB-\'h is di\'isible by D if kB-\-a-\-hd is divisible. 
W. Mantel and G. A. Oskamp'^ proved that, to test the di\isibility of a 
number to any base by a prime, the value of the coefficient required to 
eliminate one, two, . . . digits on subtraction is periodic. Also the number of 
terms of the period equals the length of the period of the periodic fraction 
arising on division by the same prime. 
G. Dostor-^'' noted that \{)t-\-u is divisible by any divisor a of 10A± 1 if 
t=^Au is di\dsible by a. [A case of Liljevalch^-''.] 
Hocevar^^ noted that if N, wTitten to base a, is separated into groups 
Gi, (x2, . . . each of q digits, N is di\'isible by a factor of a'+l if Gi — G2+G3 
- ... is divisible. Thus, for a = 2, g = 4, A'' = 104533, or 11001100001010101 
to base 2 is divisible by 17 since 0101-0101 + 1000-1001 + 1 = 0. 
J. Delboeuf^° stated that if p, q are such that pa-\-qh is a multiple of Z) and 
if N = Aa.-\-B^ is a multiple of Z) = aa + 6/3, then pA+qB is a multiple of Z). 
E. Catalan {ibid., p. 508) stated and proved the preceding test in the 
following form: If a, h and also a', h' are relatively prime, and 
iV = aa'+66', Nx = Aa-\-Bh, Nx' = A'a'+B'h', 
then AA'-\-BB' is a multiple of A^ (and a sum of 2 squares if N is). 
"Tidsskrift for Math., (3^, 5, 1875, 177-180. «*Tidsskrift for Math., (3), 6, 1876, 15-19. 
"Zeitschrift Math. Phys., 21, 1876, 365-6. »'Zeitschrift Math. Phys., 21, 1876, 366-370. 
^"''Mat. Sbornik (Math. Soc. Moscow), 8, 1876, I, 501-5. 
"Nieuw Archief voor Wiskunde. Amsterdam, 4, 1878, 57-9, 83-94. 
"-^.Ajchiv Math. Phys., 63, 1879, 221-4. 
"Zur Lehre von der Teilbarkeit. . ., Prog. Imisbruck, 1881. 
"La Revue Scientifique de France, (3), 38, 1886, 377-8. 
