CHAP. XH] CEITEKIA FOR DIVISIBILITY. 339 a=F&/n+c/n2=p ... is divisible by 1 On =±=1, with a like test for 10n=±=3 (replacing 1/n by 3/n), and deduced the usual tests for 9, 11, 7, 13, etc. A. L. Crelle16 noted that to test xmAm+... +xtA+XQ for the divisor s we may select any integer n prime to s, take r^nA (mod 5), and test ^mrm+nxm^rm-l+ ... +nmx0 for the divisor s. For example, if A = 10, 8=7, 103== 1 (mod 7), so that XQXi+x2 ... =*=xw is to be tested for the divisor 7, where XQ, .. .are the three-digit components of the proposed number from right to left. Similarly for s= 9, 11, 13, 17, 19. A. Transon16 gave a test for the divisibility of a number by any divisor of 100-n±l. A. Niegemann17 noted that 354578385 is divisible by 7 since 35457+ 2 X 8385 is divisible by 7. In general if the number formed by the last m digits of N is multiplied by k, and the product is added to the number derived from N by suppressing those digits, then N is divisible by d if the resulting sum is divisible by d. Here k(Q<k<d) is chosen so that KTfc 1 is divisible by d. Thus k = 2 if m=4, d=7. Many of the subsequent papers are listed at the end of the chapter. H. Wilbraham18 considered the exponent p to which 10 belongs modulo ra, where m is not divisible by 2 or 5. Then the decimal for 1/m has a period of p digits. If any number N be marked off into periods of p digits each, beginning with units, so that JVr = a1+10pa2+102pa3+. . ., then aj+a2+...ss.ZV (mod m), and N is divisible by m if and only if 81+02+ is divisible by m. E. B. Elliott19 let 10p=MD+rp. Thus N = 10% +... +10rc1+n0 is divisible by D if N=SnJ-MZ>+Sn/y is divisible by D. The values of the r's are tabulated for D = 3, 7, 8, 9, 11, 13, 17. A. Zbikowski20 noted that N=*a+Wk is divisible by 7 if k2a is divisible by 7. If 6 is of the form 10n+1, N = a-{- 10k is divisible by 5 if kna is divisible by 5; this holds also if 5 is replaced by a divisor of a number 10n+1. V. Zeipel21 tests for a divisor 6 by use of rib = 10d+l. Then 10a2+a! is divisible by b if a2 aid is divisible by 6. J. C. Dupain22 noted, for use when division by p 1 is easy, that N=(p l)Q+R is divisible by p if R Q is divisible by p. F. Folie23 proved that if a, c are such that ak'^ck = mp then AB-\-C is divisible by the prime p = aB+c if Ak'^Ck = mfp, provided a, c, /c, k1 are 16Jour. fur Math., 27, 1844, 12.r)~136. "Nouv. Ann. Math., 4, 1845, 173-4 (cf. 81-82 by 0. R.). 17Entwickelung u. Begriindung neuer Gesetze uber die Theilbarkeit der Zahlen. Jahresber. Kath. Gym. Koln, 1847-8. "Cambridge and Dublin Math. Jour., 6, 1851, 32. 19Thc Math. Monthly (cd. Runkle), 1, 1859, 45-49. "Bull. ac. ec. St. P6tersbourg, (3), 3, 1861, 151-3; Melanges math. astr. ac. St. PStersbourg, 3, 1859-66, 312. 216fversigt finska vetcnak. forhandl., Stockholm, 18, 1861, 425-432. "Nouv. Ann. Math., (2), 6, 1867, 368-9. 23M6m. Soc. Sc. Lidge, (2), 3, 1873, 85-96.