Chap. XII] CRITERIA FOR DIVISIBILITY. 341 
Noel (ibid., 378-9) gave tests for divisors 11, 13, 17,. . ., 43. 
Bougon {ibid., 508) gave several tests for the divisor 7. For example, 
a number is divisible by 7 if the quadruple of the number of its tens dimin- 
ished by the units digit is divisible by 7, as 1883 since 188-4 — 3 = 749 is 
divisible by 7. J. Heilmann (ibid., 187) gave a test for the divisor 7. 
P. Breton and Schobbens {ibid., 444-5) gave tests for the divisor 13. 
S. Dickstein^^ gave a rule to reduce the question of the divisibility of a 
number to any base by another to that for a smaller number. 
A. Loir^2 gave a rule to test the divisibihty of N, having the units digit a, 
by a prime P. From {N — a)/10, subtract the product of a by the number, 
say (mP — 1)/10, of tens in such a multiple mP of P that the units digit is 1. 
To the difference obtained apply the same operation, etc., until we exhaust 
N. If the final difference be P or 0, N is divisible by P. 
R. Tucker^^ started with a number N, say 5443, cut off the last digit 3 
and defined ^2 = 544 — 2-3 = 538, ^3 = 53 — 2-8, etc. If any one of the ^^'s is 
divisible by 7, N is divisible by 7. R. W. D. Christie (p. 247) extended the 
test to the divisors 11, 13, 17, 37, the respective multiphers being 1, 9, 5, 11, 
provided always the number tested ends with 1, 3, 7 or 9. 
R. Perrin^^ would find the minimum residue of N modulo p as follows. 
Decompose N, written to base x, into any series of digits, each with any 
number of digits, say A, Bi, Cj,. . ., where Bi has i digits. Let p be any 
integer prime to x and find qi so that qiX^^ 1 (mod p). Let a be any one of 
the integers prime to p and numerically <p/2. Let j8 be the ith integer 
following a in that one of the series containing a which are defined thus: 
as the first series take the residues modulo p of 1, g, g^, . . . ; as the second 
series take the products of the preceding residues by any new integer prime 
to p; etc. Let y be the jth integer following /S in the same series, etc. 
Then N' = Aa-\-BS-{-Cjy-\-... is or is not divisible by p according as 
A'' is or not. By repetitions of the process, we get the minimum residue 
of N modulo p. The special case A-{-Biqi, with p a prime, is due to Loir.^^ 
Dietrichkeit^^ would test Z = \Ok-\-a for the divisor n by testing k — xa, 
where 10a:+l is some multiple of n. To test Z (pp. 316-7) for the divisor 
7, test the sum of the products of the units digit, tens digit, ... by 1, 3, 2, 6, 
4, 5, taken in cyclic order beginning with any term (the remainders on con- 
verting 1/7 into a decimal fraction) . Similarly for 1/n, when n is prime to 10. 
J. Pontes^ ^ would test N for a divisor M by using a number <iV and 
= N (mod M), found as follows. For the base B, let q be the absolutely 
least residue of B"" modulo M. Commencing at the right, decompose N 
into sets of m digits, as X,„, . . . , a^, and set f{x)=a^x"'+^jn^"'~^-\- . . . +X;;,, 
whence N=f{B'^). By expanding N=f{q+M^), we see that f{q) is the 
desired number < N and = N (mod M) . 
S. Levanen^^ gave a table showing the exponent to which 10 belongs for 
siLemberg Museum (Polish), 1886. "Comptes Rendus Paris, 106, 1888, 1070-1; errata, 1194. 
^'Nature, 40, 1889, 115-6. 34 Assoc, franp. avanc. sc, 18, 1889, II, 24-38. 
'^Zeitschr. Math. Phys., 36, 1891, 64. 3«Comptes Rendus Paris, 115, 1892, 1259-61. 
"Ofversigt af finska vetenskaps-soc. forhandUngar, 34, 1892, 109-162. Cf . Jahrbuch Fortschr. 
Math., 24, 1892, 164-5. 
