CHAP, xi] GREATEST COMMON DIVISOE. 333 Q by (x-p-I).. . (x—2p) and call the quotient Qf and remainder Rf. Then must Xj>R'-\-X2vQf^Q and hence XPR'^Q (mod p"). Thus if ^ is the exponent of the highest power of p dividing the coefficients of R', we have M=M2+1. In general, if JUA and \h__l are the exponents of the highest powers of p dividing the coefficients of the remainder R(k~l) and Z(Jb_1)p identically, then /i^Mfc+X*-i- Finally, if l = [m/p], M^XZ. The extension to several variables is said to present difficulties. [For simpler methods, see Hensel44 and Borel.48] It is noted (p. 323) that are identically divisible by pn. It is conjectured (p. 328) that/(rc)/JV represents an infinitude of primes when f(x) is irreducible. E. Cesaro37 and J. J. Sylvester^8 proved that the probability that two numbers taken at random from 1, . . ., nbe relatively prime is 6/?r2 asymptotically. L. Gegenbauer39 gave 16 sums involving the g. c. d. of several integers and deduced 37 asymptotic theorems such as the fact that the square of the g. c. d- of four integers has the mean value 15/?r2. He gave the mean of the fcth power of the g. c. d. of r integers. J. Neuberg390 noted that, if two numbers be selected at random from 1, . . . , N, the probability that their sum is prime to N is k = <t>(N) or k/(N—l) according as N is odd or even. T. J. Stieltjes,40 starting with a set of n integers, replaced two of them by their g. c. d. and 1. c. m., repeated the same operation on the new set, etc. Finally, we get a set such that one number of every pair divides the other. Such a reduced set is unique. The 1. c. m. of a, . . ., I can be expressed (pp. 14-16) as a product a' . . .V of relatively prime factors dividing a, . . . , lj respectively. The 1. c. m. (or g. c. d.) of a, 6, . . . , Z equals the quotient of P = ab . . .1 by the g. c. d. (or 1. c. m.) of P/a, P/b, . . . , P/l E. Lucas41 gave theorems on g. c. d. and 1. c. m. L. Gegenbauer41'1 considered in connection with the theory of primes, the g. c. d. of r numbers with specified properties. J. Hacks42 expressed the g. c. d. of m and n in the forms where € = 0 or 1 according as m, n are both or not both even. J. Hammond43 considered arbitrary functions / and F of p and a, such "Mathcsis, 1, 1881, 184; Johns Hopkins Univ. Circ., 2, 1882-3, 85. a8Johns Hopkins Univ. Circ., 2, 1883, 45; Comptes Rendus Paris, 96, 1883, 409; Coll. Papers, 3, 675; 4, 86. "SitzungBbcrichtc Ak. Wins. Wien (Math.) 92, 1885, II, 1290-1306. ""Math. Quest. Kduc. Times, 50, 1889, 113-4. 40fiur la thdoric dos nornbrca, Annalca dc la, fac. dca sciences de Toulouse, 4, 1890, final paper.