CHAP, xii] CRITERIA FOR DIVISIBILITY. 343 To test N= a0+ai#+ . . . + anJ5nfor the divisor D prime to B, determine d and x so that Bd — Dx+l. Multiply this equation by aQ and subtract from N. Thus N' = aQd+(al+a2B+ . . . Hence N is divisible by D if and only if Nr is divisible by D. Now, N' is derived from N by supressing the units digit a0 and adding to the result the product aQd. Next operate with N' as we did with N. J. Malengreau46 would test N for a factor q prime to 10 by seeking a multiple 11 ... 1 (to m digits) of q, then an exponent t such that the number of digits of lO'-AT is a multiple of m. From each set of m digits of W-N subtract the nearest multiple of 1 ... 1 (torn digits) . The sum of the residues is divisible by q if and only if N is divisible by q. G. Loria47 proved that N = a^+ga\+ . . . +gkak is divisible by a if and only if a divides the sum a0+ . . .+akoi the digits of N written to a base g of the form ka + 1 ; or if a divides a0 — ai + a2 — . . . when the base g is of the form ka—1. Taking g = 10m, we have the test, in Gelin's Arithm£tique, in terms of groups of m digits. We may select m to be ^<X°) co* a number such that 10m=t= l has the factor a. In place of a0+ai+ . . . when^ = 10m, we may employ where X = l, 2 or 5, and p is determined by 10p/X^l (mod a). Taking a = 7, 13, 17, 19, 23, special tests for divisors are obtained. G. Loria48 proved that, if a0, a1? . . . are successive sets of t digits of N, counted from the right, and cr = a0=fcai-|-a2=ta3+ . . ., then so that a factor of 10'=Fl divides N if and only if it divides a. A. Tagiuri49 extended the last result to any base g. We have # = 00+0^+ . . . =NQm+gmNlrn+g2mN2m+ . . . if Npm = af>m+apm+1g+ . . . +apm+m_1^m~1. Hence, if gm= ±1 (mod a), N=NGm±Nlm+N2m*= . . . (mod a). L. Ripert60 noted that lOD+u is divisible by 105-K if Di-bu is divisible, and gave many tests for small divisors. G. Biase61 derived tests that Wd+u has the factor 7 or 19 from = 2u-d (mod 7), 2(Wd+u) = 2u+d (mod 19). O. Meissner52 reported on certain tests cited above. "Matheeis, (3), 1, 1901, 197-8. "Rendiconti Accad. Lincei (Math.), (5), 10, 1901, sem. 2, 150-8. Mathesis, (3), 2, 1902, 33-39. 48I1 Boll. Matematica Gior. Sc.-Didat., Bologna, 1, 1902. Cf. A. Bindoni, ibid., 4, 1905, 87. 48Periodico di Mat., 18, 1903, 43-45. ML»enfleignement math., 6, 1904, 40-46. B1I1 Boll. Matematica Gior. Sc.-Didat., Bologna, 4, 1905, 92-6. "Math. Naturw. Blatter, 3, 1906, 97-99.