338 History of the Theory of Numbers. [Chap, xii 
D'Alembert^ noted that if N = A-10'"+B-W-\-. ..+E is divisible by 
10-6, then Ab"'+Bb"-\- . ..+E is divisible by 10-6; if A' is divisible by 
10+6, then A(-6)'"+B(-6)"+ . . . +^ is divisible by 10+6. The case 
6 = 1 gives the test for divisibiUty by 9 or 11. By separating A'' into parts 
each with an even number of digits, N = A-10"'+ . . . +E, where m, . . .are 
even; then if A^ is di\'isible by 100-6, Ah"*^^ -\- . . . + E is divisible by 
100-6. 
De Fontenelle^ gave a test for divisibility by 7 which is equivalent to 
the case 6 = 3 of D'Alembert; to test 3976 multiply the first digit by 3 and 
add to the second digit; it remains to test 1876. For proof see F. Sanvitali, 
Hist. Literariae Italiae, vol. 6, and Castelvetri.^ 
G. W. Kraft^ gave the same test as Pascal for the factor 7. 
J. A. A. Castelvetri^ gave the test for 99: Separate the digits in pairs, 
add the two-digit components, and see if the sum is a multiple of 99. For 
999 use triples of digits. 
Castelvetri^ tested 1375, for example, for the factor 11 by noting that 
13+75 = 88 is divisible by 11. If the resulting sum be composed of more 
than two digits, pair them, add and repeat. To test for the factor 111, 
separate the digits into triples and add. The proof follows from the fact 
that lO-*" has the remainder 1 when divided by 11. 
J. L. Lagrange^° modified the method of Pascal by using the least 
residue modulo A (between — .4/2 andyl/2) in place of the positive residue. 
He noted that if a number is written to any base a its remainder on division 
by a — 1 is the same as for the sum of its digits. 
J. D. Gergonne^^ noted that on di\dding iV = Ao+Ai6"*+A26^'"+ . . ., 
written to base 6, by a di\'isor of 6'" — 1, the remainder is the same as on 
dividing the sum A0+A1+A2+ ... of its sets of m digits. Similarly for 
6'"+l and A0-A1+A2-A3+ • • •• 
C. J. D. HilP^ gave rules for abbre\dating the testing for a prime factor 
p, for p<300 and certain larger primes. 
C. F. Liljevalch^^a ^^^^^ ^^^^^ jf lO^a-/? is di\^sible by p then a- 10^6 
will be a multiple of p if and only if aa — /36 is a multiple of p. 
J. ]\I. Argardh" used Hill's symbols, treating divisors 7, 17, 27, 1429. 
F. D. Herter^^ noted that a + 106+100c+ ... is divisible by 10n±l if 
'Manuscript R. 240* 6 (8°), Bibl. Inst. France, 21, ff. 316-330, Sur une propri^t^ des nombres. 
•Histoire Acad. Paris, ann^e 1728, 51-3. 'Comm. Ac. Sc. Petrop, 7, ad annos 1734-5, p. 41. 
»De Bononiensi Scientiarum et Artium Institute atque Academia Comm., 4, 1757; commen- 
tarii, 113-139; opuscula, 242-260. 
•De Bononiensi Scientiarum et Artium Institute atque Academia Comm., vol. 5, 1767, part 1, 
pp. 134-144; part 2, 108-119. 
"Lemons 6\6m. sur les math, donn^es k I'^cole normale en 1795, Jour, de I'^cole polytechnique, 
vols. 7, 8, 1812, 194-9; OemTes, 7, pp. 203-8. 
"Annales de math, (ed., Gergonne), 5, 1814-5, 170-2. 
"Jour, fur Math., 11, 1834, 251-261; 12, 1834, 355. Also, De factoribua numerorum com- 
positonim dignoscendis, Lund, 1838. 
"<»De factoribus numerorum compositorura dignoscendis, Lund, 1838. 
"De residuis ex divisione. . ., Diss. Lund, 1839. 
"Ueber die Kennzeichen der Theiler einer Zahl, Progr. Berlin, 1844. 
