CHAP, xi] GREATEST COMMON DIVISOK. 335 mn is the mean. When m and n increase indefinitely, the mean becomes 6/O2Z2). The case t = l 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. Dintzl50 proved that the g. c. d. A (a, . . ., e) is a linear function of a, . . . , e, and reproduced the proof of Lebesgue's35 formula as given in Merten's Vorlesungen liber Zahlentheorie and by de Jough.61 A-. Pichler,500 given the 1. c. m. or g. c. d. of two numbers and one of them, found values of the other number. J. C. Kluyver52 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's53 function the somewhat different form w W. Sierpinski64 stated that the probability that two integers ^>n are relatively prime is * „ contrary to Bachmann, Analyt. Zahlentheorie, 1894, 430. G. Darbi55 noted that if a = (a, N) is the g. c. d. of a, N, 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. Gelin56 noted that the product of n numbers equals ab, where a is the 1. c. m. of their products r at a time, and 6 is the g. c. d of their products n— rat a time. B. F. Yanney57 considered the greatest common divisors A, A, ... of ai, . . . , an in sets of k, and their 1. c. m.'s I/i, L2) . . . . Then H A Lf-1 £(<»!... an)c ^ TLDf-1!*, b = (?) , c = .'-I i-l W The limits coincide if k = 2. The products have a single term if k — n. P. Bachmann58 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 $(n)- If 5 = n/d occurs to the rth power, but not to the fc+ l)th power in n, set nl = n/br. From ^ build 5l as before, etc. Then N = 65i52 .... 60Zeitschrift fur das Realschulwesen, Wien, 27. 1902, 654-9, 722. 6oalbid., 26, 1901, 331-8. 61Nieuw Archief voor Wiskunde, (2), 5, 1901, 262-7. 62K. Ak. Wetenschappen Amsterdam, Proceedings of the Section of Sciences, 5, II 1903, 658- 662. (Versl. Ak. Wet., 11, 1903, 782-6.) »Jour. fur Math., 102, 1888, 9-19. "Wiadomosci Mat., Warsaw, 11, 1907, 77-80. 65Giornale di Mat.', 46, 1908, 20-30. 66I1 Pitagora, Palermo, 16, 1909-10, 26-27. "Amer. Math. Monthly, 19, 1912, 4-6.