Chap. XI] GREATEST COMMON DiVISOR. 335 
mn is the mean. When m and n increase indefinitely, the mean becomes 
Q/iirH"^). The case ^=1 gives the probability that two arbitrarily chosen 
integers are relatively prime; the proof in Dirichlet's Zahlentheorie fails to 
establish the existence of the probability. 
E. DintzP° proved that the g. c. d. A(a, . . ., e) is a linear function of 
a, . . . , e, and reproduced the proof of Lebesgue's^^ formula as given in 
Merten's Vorlesungen iiber Zahlentheorie and by de Jough.^^ 
A. Pichler,^°" given the 1. c. m. or g. c. d. of two numbers and one of them, 
found values of the other number. 
J. C. Kluyver^^ constructed several functions z (involving infinite series 
or definite integrals) which for positive integral values of the two real 
variables equals their g. c. d. He gave to Stern's^^ function the somewhat 
different form [;r] / \ 
W. Sierpinski^^ stated that the probability that two integers ^n are 
relatively prime is . „ p -,2 
contrary to Bachmann, Analyt. Zahlentheorie, 1894, 430. 
G. Darbi^^ noted that if a = (a, N) is the g. c. d. of a, N, 
(iV,abc...)=a(6,^)(c,^(^_^/J 
and gave a method of finding the g. c. d. and 1. c. m. of rational fractions 
without bringing them to a common denominator. 
E. Gelin^® noted that the product of n numbers equals ah, where a is 
the 1. c. m. of their products r at a time, and h is the g. c. d of their products 
n — r at a time. 
B. F. Yanney^'^ considered the greatest common divisors Di, D2, ... of 
tti, . . . , a„ in sets of k, and their 1. c. m.'s Li, L2, . . . . Then 
HA Lt' ^ (ai . . . «n)^ ^ n D ^-^L„ 5 = (^y c = (^~ I) . 
The limits coincide ii k = 2. The products have a single term iik = n. 
P. Bachmann^^ showed how to find the number N obtained by ridding 
a given number n of its multiple prime factors. Let d be the g. c. d. of n 
and 0(n). If d = n/d occurs to the rth power, but not to the (r+l)th power 
in n, set ni = n/5^ From rii build di as before, etc. Then N = 86182 .... 
"Zeitschrift fiir das Realschulwesen, Wien, 27. 1902, 654-9, 722. 
6o«76id., 26, 1901, 331-8. 
"Nieuw Archief voor Wiskunde, (2), 5, 1901, 262-7. 
^^K. Ak. Wetenschappen Amsterdam, Proceedings of the Section of Sciences, 5, II 1903, 658- 
662. (Versl. Ak. Wet., 11, 1903, 782-6.) 
"Jour, flir Math., 102, 1888, 9-19. 
"Wiadomosci Mat., Warsaw, 11, 1907, 77-80. 
^^Giornale di Mat., 46, 1908, 20-30. 
S6I1 Pitagora, Palermo, 16, 1909-10, 26-27. 
"Amer. Math. Monthly, 19, 1912, 4-6. 
6«Archiv Math. Phys., (3), 19, 1912, 283-5. 
