CHAP, vi] PERIODIC DECIMAL FRACTIONS. 169
Karl Broda68 considered a periodic decimal fraction F having an even number r of digits in the period and a number w of p digits preceding the period. Let x be the first half of the period, y the second half. Then
JLj __ 5 __ I y i ' x i = P I m m+r m+2r"rw+3r ' * ' m
__ __ =
10m 10m+r 10m+2r"r10w+3r ' * ' 10m 10m(102r~l)
_9(pl(f+x+p)+a
— l)/9 = a. . .a (to r terms). The first paper treated the case 0, and gave the generalization to base a in place of 10:
x . y . x a+(a — l)s ,r . arl

The case a=a — 1 shows that a purely periodic fraction to the base a equals (z+l)/(ar+l) if the sum of the half periods has all its digits (to base a) equal to a— 1. Returning to the base 10, and taking N= 9(10r+l), Z = 9z+a, where each digit of 3 is ^a, we see that Z/N equals a decimal fraction in which x is the first half of the period of r digits, while the second hah" is such that the sum of corresponding digits in it and x is a. If R is the remainder after r digits of the period have been obtained, R+Z=a (10r+l)«
C. G. Reuschle69 gave tables which serve to find numbers belonging to a given exponent < 100 with respect to a given prime modulus < 1000.
P. Mansion70 gave a detailed proof that, if n is prime to 2, 3, 5, and if the period for 1/n has n— 1 digits, the sum of corresponding digits in the half periods is 9.
T. Muir71 proved that, If p is a prime, either of
Nx= 1 (mod p1), 2V*"n=l (mod p°+n)
follows from the other. If Xi is the least positive integer x for which the first holds and if p8 is the highest power of p dividing NXl — 1, then Xipn is the least positive integer y for which Ny= 1 (mod p*+n). Hence the known theorem: If N=Hpini} where pi, p2, . . . are distinct primes, and if the period for I/ pi has mt digits, and if p* is the highest power of Pi dividing 10™* — 1, the number of digits in the period for l/N is the 1. c. m. of the mip"^*. He asked if 6 = 1 when p>3, as affirmed by Shanks.65.
Mansion's proof (ibid., 5, 1876, 33) by use of periodic decimals of the generalized Fermat theorem is quoted under that topic.
D. M. Sensenig72 noted that a prime p5^2, 5, divides N if it divides the sum of the digits of N taken in sets of as many figures each as there are digits in the period for I/ p.
"Archiv Math. Phys., 56, 1874, 85£8; 57, 1875, 297301.
"Tafeln complexer Primzahlen, Berlin, 1875. Errata by Cunningham, Mess. Math., 46,
1916, 601.
70Nouv. Corresp. Math., 1, 18745, 812. "Messenger Math.. 4, 1875, 15. "The Analyst, Des Moines, Iowa, 3, 1876, 25.