Chap. VI] PERIODIC DECIMAL FRACTIONS. 163 
If M is divisible by p, we may take n = 1 and conclude that A^p^ differs 
from 1/p^ by an integer. If M is not divisible by p, S must be, so that n 
is divisible by p and the length of the period is pt. In general, for the denom- 
inator p^, we have n = l if M is divisible by p^~^, but in the contrary case 
n is a multiple of p'"'^. If the period for a prime p has an even number of 
digits, the sum of corresponding quotients in the two half periods is p. 
An anonymous writer^^ noted that, if we add the digits of the period of 
a circulating decimal, then add the digits of the new sum, etc., we finally 
get 9. From a number subtract that obtained by reversing its digits; add 
the digits of the difference; repeat for the sum, etc.; we get 9. 
Bredow^^ gave the periods for a/p, where p is a prime or power of a 
prime between 100 and 200. He gave certain factors of 10" — 1 for w = 6-10, 
12-16, 18, 21, 22, 28, 33, 35, 41, 44, 46, 58, 60, 96. 
E. Midy" noted that, if a"", a"', . . . are the least powers of a which, 
diminished by unity, give remainders divisible by q^, qi''', . . . , respectively 
{q, qi,... being distinct primes), and if the quotients are not divisible by 
q, qi,. . ., respectively, and if t is the 1. c. m. of n, ni, . . ., then a belongs to 
the exponent t modulo p = q^qi^' . . . , and a' — 1 is divisible by q only h times. 
Let the period of the pure decimal fraction for a/h have 2n digits. If 
h is prime to 10" — 1, the sum of corresponding digits in the half periods is 
always 9, and the sum of corresponding remainders is h. Next, let 6 and 
10" — 1 have d>l as their g. c. d. and set h' = h/d. Let a„ be the nth re- 
mainder in finding the decimal fraction. Then a+a„ = 6'A:, ai+a„+i = 6'/ci, 
etc. The sums q-\-qn, 5i+g„+i, ... of corresponding digits in the half 
periods equal 
{\{)k-k^)/d, il0k,-k2)/d,.. ., {10k,_r-k)/d. 
Similar results hold when the period of mn digits is divided into n parts of 
m digits each. For example, in the period 
002481389578163771712158808933 
for 1/403, the two halves are not complementary (10^^ — 1 being divisible 
by 31); for i = l, 2, 3, the sum of the digits of rank i, i-\-3, i+6, . . ., i+27 
is always 45, while the corresponding sums of the remainders are 2015. 
N. Druckenmiiller^^" noted that any fraction can be expressed as a/x-\- 
ai/x^-l- .... 
J. Westerberg^^ gave in 1838 factors of 10"± 1 for nS 15. 
G. R. Perkins^^ considered the remainder r^ when N'^ is divided by P, 
and the quotient q in Nrj._i = Pqx-\-rj.. If Tk'^P—l, there are 2k terms in 
the period of remainders, and 
rk+x+r^ = P, qk+x+qx = N-l. 
[These results relate to 1/P written to the base N.] 
^^Polytechnisches Journal (ed., J. G. Dingier), Stuttgart, 34, 1829, 68; extract from Mechanics' 
Magazine, N. 313, p. 411. 
^*Von den Perioden der Decimalbriiche, Progr., Oels, 1834. 
'^'De quelques propriet^s des nombres et des fractions d^cimales p^riodiques, Nantes, 1836,21 pp. 
""T.heorie der Kettenreihen . . ., Trier, 1837. 
28See Chapter on Perfect Numbers."* 
2»Amer. Jour. Sc. Arts, 40, 1841, 112-7. 
