Chap. VI] PeEIODIC DECIMAL FRACTIONS. 171 
place of the product ap, /S the digit in the tens place. Write the digit b 
to the left of digit a to form the last two digits of the required number P. 
The number c in the units place in 6p+j8 is written to the left of digit h in 
P. To cp add the digit in the tens place of bp and place the unit digit of 
the sum to the left of c in P. The process stops with the kth. digit t if the 
next digit would give a. Then P = t. . . cba and its products by k integers or 
fractions has the same k digits in the same cyclic order. For a = 2, p = 3, 
we get A; = 28 and see that P is the period of 2/27, and the k multipliers 
are m/2, m = l,. . ., 28. [To have an example simpler than the author's, 
take a = 7, p = 5; then P = 142857, the period of 1/7; the multipliers are 
1, . . . , 6.] For proof, we have 
P = 10''-H-\- . . .-\-10h+10b+a, pP = 10''-'a+10''-H+ . . .+10c-\-h, 
pP = 10^-a+^, 10^ = 10^' 
so that P is the period with k digits for a/{lOp — l). 
E. LucasS2 gave the prime factors of lO'^^l, 10'^±1, lO^^^l, 10^^+1, 
10^^+1, communicated to him by W. Loof, with the remark that (10^^ — 1)/9 
has no prime factor < 3035479. Lucas gave the factors of 10^^+1. 
J. W. L. Glaisher^^ proved his^^ earlier statements, repeated his" earher 
remarks, and noted that, if g is a prime such that the period for 1/q has q — 1 
digits, the products of the period for 1/q by 1, 2, . . ., g — 1 have the same 
digits in the same cyclic order. This property, well known for q = 7, holds 
also for g = 17, 19, 23, 29, 47, 59, 61, 97 and for q = 7\ 
0. Schlomilch^^ stated that, to find every N for which the period for 
1/N has 2k digits such that the sum of the sth and (fc+s)th digits is 9 for 
s = 1, . . . , A;, we must take an integer iV= (10*^+l)/r; then the first k digits 
of the period are the k digits of T — 1. 
C. A. Laisant^^ extended his investigations with Beaujeux^^'^^ and gave a 
summary of known properties of periodic fractions; also his^^ process to 
find the period of simple periodic fractions without making divisions. 
V. Bouniakowsky^^ noted that the property of the period of 1/N, 
observed by Schlomilch^^ for iV = 7, 11, 13, 77, 91, 143, holds also for the pe- 
riods of /c/iV, for A; = iV — 1 and (iV — 1)/2, with the same values of AT. Consider 
the decimal fraction Q.yiy2- ■ ■ with ym — ym-i+ym-2 (mod 9), replacing any 
residue zero by 9, and taking yi > 0, 1/2 > 0- The fraction is purely periodic 
and is either 0.9 or 0.33696639 or has the same digits permuted cyclically, 
or else has a period of 24 digits and begins with 1, 1 or 2, 2 or 4, 4, or has the 
same 24 digits permuted cyclically or by the interchange of the two halves 
s^Nouv. Corresp. Math., 5, 1879, 138-9. 
8'Nature, 19, 1879, 208-9. 
s^Zeitschrift Math. Phys., 25, 1880, 416. 
8*M6m. Soc. Sc. Phys. et Nat. de Bordeaux, (2), 3, 1880, 213-34. 
86Le8 Mondes, 19, 1869, 331. 
"BuU. Acad. Sc. St. P^tersboiirg, 27, 1881, 362-9. 
