Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 273 
C. Szily^^ noted that no prime >2a divides 
?M) • 
and specified the intervals in which its prime factors occur. 
F. Morley^^ proved that, if p = 2n+l is a prime, (2„")-(-l)"2*" is 
divisible by p^ ii p>S. That it is divisible by p^ was stated as an exercise 
in Mathews' Theory of Numbers, 1892, p. 318, Ex. 16. 
L. E. Dickson^° extended Rummer's''^ results to a multinomial coefficient 
M and noted the useful corollary that it is not divisible by a given prime p 
if and only if the partition of m into nii,..., nit arises by the separate 
partition of each digit of m written to the base p into the corresponding 
digits oi TUi, . . . , rrit. In this case he proved that 
^= n .1),''" .,, (mod p), m, = ao''Y-{- . . . +a,^''K 
This also follows from (2) and from 
(xi+ . . . +xtr= {x,+ ... +x,)"»(xiP+ . . . +x,o"»-i . . . (0^1^''+ . . . +x/'ro 
(mod p). 
F. Mertens^^ considered a prime p^n, the highest powers p" and 2" of 
p and 2 which are ^n, and set n„ = [n/2"]. Then nl-^ {niln2l. . .nj} is 
divisible by Up'", where p ranges over all the primes p. 
J. W. L. Glaisher^^ gave Dickson's^" result for the case of binomial 
coefficients. He considered (349-60) their residues modulo p"', and proved 
(pp. 361-6) that if {n)r denotes the number of combinations of n things r 
at a time, 'Z{n)r^(j)k (mod p), where p is any prime, n any integer =j 
(mod p — l), while the summation extends over all positive integers r, 
f"^n, r=k (mod p — l), and j, k are any of the integers 1,. . ., p — l. He 
evaluated S[(?^)r-^p] when r is any number divisible by p — l, and (n)^ is 
divisible by p, distinguishing three cases to obtain simple results. 
Dickson^^ generalized Glaisher's^^ theorem to multinomial coefficients: 
Let k be that one of the numbers 1, 2, . . ., p — l to which m is congruent 
modulo p — l, and let ki,..., kt be fixed numbers of that set such that 
ki-\- ■ ■ ■ -\-kt=k (mod p — l). Then if p is a prime, 
where , . , m 
(mi, . . . , nit) = J ; 
The second of the two proofs given is much the simpler. 
ssNouv. Ann. Math., (3), 12, 1893, Exercices, p. 52.* Proof, (4), 16, 1916, 39-42. 
s^Annals of Math., 9, 1895, 168-170. 
^"Ibid., (1), 11, 1896-7. 75-6: Quart. Jour. Math., 33, 1902, 378-384. 
siSitzungsber. Ak. Wiss. Wien (Math.), 106, lla, 1897, 255-6. 
'2Quar. Jour. Math., 30, 1899, 150-6, 349-366. 
o^Ibid., 33, 1902, 381-4. 
