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338                   HISTORY or THE THEORY OF NUMBERS.            [CHAP, xii
D'Alembert6 noted that if N=A-10m+B-10n+.. .+E is divisible by 10 ~b, then Abm+Bbn+.. ,+E is divisible by 106; if N is divisible by 10+b, then A(-b)m+B(-b)n+.. .+E is divisible by 10+6. The case 6 = 1 gives the test for divisibility by 9 or 11. By separating N into parts each with an even number of digits, ^ = ^-10"*+ ... +E, where m,.. .are even; then if N is divisible by 100-6, Abm/2+... +E is divisible by 100-6.
De Fontenelle6 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.8
G. W. Kraft7 gave the same test as Pascal for the factor 7.
J. A. A. Castelvetri8 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.
Castelvetri9 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 102Jfc has the remainder 1 when divided by 11.
J. L. Lagrange10 modified the method of Pascal by using the least residue modulo A (betweenA/2 and A/2) in place of the positive residue. He noted that if a number is written to any base a its remainder on division by a1 is the same as for the sum of its digits.
J. D. Gergonne11 noted that on dividing N = AQ+A1bm+A2b2m+.. ., written to base 6, by a divisor of 6m1, the remainder is the same as on dividing the sum A0-f Ai~+-A2+... of its sets of m digits. Similarly for 6m+1 and A0-A1+A2-A^ ....
C. J. D. Hill12 gave rules for abbreviating the testing for a prime factor p, for p<300 and certain larger primes.
C. F. Liljevalch120 noted that if 10na-/3 is divisible by p then a-10n6 will be a multiple of p if and only if aafib is a multiple of p.
3. M. Argardh13 used Hill's symbols, treating divisors 7, 17, 27, 1429.
F. D. Herter14 noted that a+106+100c+... is divisible by 10nl if
5Manuscript R. 240* 6 (8), Bibl. Inst. France, 21, ff. 316-330, Sur une propri<t<$ des nombres. "Histoire Acad. Paris, anne*e 1728, 51-3. 7Comm. Ac. Sc. Petrop, 7, ad annos 1734-5, p. 41. 8De 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. 10Lecon3 616m. sur les math. donne*es d l^cole normale en 1795, Jour, de 1'e'cole polytechnique,
vols. 7, 8, 1812, 194-9; Oeuvres, 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 factoribus numerorum com-
positorum dignoscendis, Lund, 1838.
12aDe factoribus numerorum compositorum dignoscendis, Lund, 1838. 13De residuis ex divisione..., Diss. Lund, 1839. 14Ueber die Kennzeichen der Theiler emer Zahl. Proerr. Berlin. 1K44.