Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 271 
Hence, by Legendre's formula (1), 
ip-l)N = A-\-B-y-ei-{A-a)-{B-P), a=Sa„ ^ = 26^, 7=Sc,. 
Insert the value of a+/3 obtained by adding equations (3). Thus 
A. Genocchi'^^ proved that, if m is the sum of n integers a,h,...,k, each 
divisible by p — l, and if m<p" — 1, then m\-i- {a\bl. . .k\} is divisible by the 
prime p. 
J. Wolstenholme^^ proved that f"ll) = l(mod n^) if n is a prime > 3. 
H. Anton^ (303-6) proved that if n = vp+a, r = wp+h, where a, h, v, w 
are all less than the prime p, 
according as a^ 6 or a < 6. 
M. Jenkins^^" considered for an odd prime p the sum 
"^^ ^\mr+k{p-l)J' 
extended over all the integers k between nr/(p — l) and —mr/{p — l), in- 
clusive, and proved that (Tr=o'p (mod p) if the g. c. d. of r, p — 1 equals that 
of p, p-1. 
E. Catalan^^ noted that C'^_}) = l(mod p), if p is a prime. 
Ch. Hermite^^ proved by use of roots of unity that the odd prime p divides 
/2n+l\ , /2n+l\ , /2n+l\ , 
{p-l) + [2p-2r[sp-3r-' 
E. Lucas'^^ noted that, if m = pmi-\-ii, n = pni+v, ii<p, v<p, and p is 
a prime. 
In general, if fxi, fJL2, ■ ■ ■ denote the residues of m and the integers contained 
in the fractions m/p, m/p^, . . . , while the v's are the residues of n, [n/p], . . . , 
e)-t;)t)- '-'^^'- 
E. Lucas'^'^ proved the preceding results and 
0-0. f ;>(-!)". (^:>0(modp), 
according as n is between and p, and p — l, or 1 and p. 
"Nouv. Ann. Math., 14, 1855, 241-3. 
"Quar. Jour. Math., 5, 1862, 35-9. For mod. w^ Math. Quest. Educ. Times, (2), 3, 1903, 33. 
"'^Math. Quest. Educ. Times, 12, 1869, 29. ^■'Nouv. Corresp. Math., 1, 1874-5, 76. 
'^Jour. fur Math., 81, 1876. 94. '^Bull. Soc. Math. France, 6, 1877-8, 52. 
"Amer. Jour. Math., 1, 1878, 229, 230. For the second, anon.« of Ch. Ill (in 1830). 
