CHAP. IX] DIVISIBILITY OF FACTOEIALS, MULTINOMIAL COEFFICIENTS. 265 is an integer, and wrote Lk for its residue modulo pk+1. Set He proved that TT< is divisible by pa, where s irt+Sj"oS^. If TV is the first one of the numbers TO, n,. . . which is <p 1, TT, is divisible by p', A. Cunningham13 proved that if z* is the highest power of the prime 2 dividing p, the number of times p is a factor of pnl is the least of the numbers n n-f+i for the various primes z. dividing p. W. Janichen14 stated and G. Szego proved that summed for the divisors d of n, where v(d) is the exponent of the highest power of p (a prime factor of n) which divides dl, f or JJL as in Ch. XIX. INTEGRAL QUOTIENTS INVOLVING FACTOEIALS. Th. Schonemann18 proved, by use of symmetric functions of pth roots of unity, that if 5 is the g. c. d. of ju, *>, . . . , ^(m 1)! - ^- He gave (p. 289) an arithmetical proof by showing that the fractions obtained by replacing d by /x, v, . . . are integers. A. Cauchy19 proved the last theorem and that (a+2&+...+nfc)-(7n-l)! , / , t ^^ ^^ - . ,/ - - = integer, (m = a+...+K). a \ . . . /c ' D. Andre"20 noted that, except when n = l, a = 4, n(n+l) . . ,(na~l) is not or is divisible by an according as a is a prime or not. E. Catalan21 found by use of elliptic functions that frn+n-1)! (2m)!(2n)! mini m!n!(m+n)! are integers, provided m, n are relatively prime in the first fraction. 13L'interm<§diaire des math., 19, 1912, 283-5. Text modified at suggestion of E. Maillet. "Archiv Math. Phys., (3), 13, 1908, 361; 24, 1916, 86-7. 18Jour. far Math., 19, 1839, 231-243. "Comptes Rendus Paris, 12, 1841, 705-7; Oeuvres, (1), 6, 109. 20Nouv. Ann. Math., (2), 11, 1872, 314. *llbid., (2), 13, 1874, 207, 253. Arith. proofs, Amer. Math. Monthly, 18, 1911, 41-3.