CHAP. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 269
M. Jenkins480 counted in two ways the arrangements of n—fyf+yg-}-... elements in </> cycles of/ letters each, y cycles of g letters, ..., where/, g, ... are distinct integers > 1, and obtained the result
C. de Polignac49 investigated at length the highest power of nl dividing (nx) \/(xl)n. Let np be the sum of the digits of n to base p. Then
(x+n)p=xp+np~k(p-l), (xri)p = xp'np-k'(p--l),
where k is the number of units "carried" in making the addition x+n, and k' the corresponding number for the multiplication x-n.
E. Schonbaum50 gave a simplified exposition of Landau's first paper.44
S. K. Maitra61 proved that (n -1) (2n -1). .. {(n- 2)n -1} is divisible by (n—1)! if and only if n is a prime.
E. Stridsberg52 gave a very elementary proof of Hermite's29 result.
E. Landau53 corrected an error in his44 proof of the result in No. Ill of his paper, no use of which had been made elsewhere.
Birkeland18 of Ch. XI noted that a product of 2pk consecutive odd integers is = l (mod 2P).
Among the proofs that binomial coefficients are integers may be cited those by:
G. W. Leibniz, Math. Schriften, pub. by C. I. Gerhardt, 7, 1863, 10?
B. Pascal, Oeuvres, 3, 1908, 278-282.
Gioachino Pessuti, Memorie di Mat. Soc. Italiana, 11, 1804, 446.
W. H. Miller, Jour, fur Math., 13, 1835, 257.
S. S. Greatheed, Cambr. Math. Jour., 1, 1839, 102, 112.
Proofs that multinomial coefficients are integers were given
C. F. Gauss, Disq. Arith., 1801, art. 41.
Lionnet, Complement des e*le*ments d'arith., Paris, 1857, 52. V. A. Lebesgue, Nouv. Ann. Math., (2), 1, 1862, 219, 254.
FACTORIALS DIVIDING THE PRODUCT OF DIFFERENCES
H. W. Segar60 noted that the product of the differ' integers is divisible by (r —l)!(r—2)!.. .2!. For t integers 1, 2,.. ., n, r+1, the theorem shows that consecutive integers is divisible by n!.
A. Cayley61 used Segar's theorem to prove that
m(m—n).. .(m—r — lri)-nr
is divisible by r! if m, n are relatively prime [a part of L Segar62 gave another proof of his theorem. ApplyinL
««Quar. Jour. Math., 33,1902,17^-9. 4fiBull. Soc. Math. France, 32, 1904, <. '"Casopis, Prag, 34, 1905, 265-300 (Bohemian). "Math. Quest. Educat. Times, (2), 12, 1907, 84-5.
MActa Math., 33, 1910, 243. 63Nouv. Ann. Math., (4), 13, 1913, 353-5.
""Messenger Math., 22, 1892-3, 59. "Messenger Math. 22, 1892-3, p. 186. Cf. j 62/6id., 23, 1893-4, 31. Results cited in 1'interme'diaire des math., 2, 1895, 132-3, 20Q; 197; 8, 1901, 145.