Chap. IX] DIVISIBILITY OF FACTORIALS, MULTINOMIAL COEFFICIENTS. 269 
M. Jenkins^^" counted in two ways the arrangements of n = 4>f-\-'yg-\- . . . 
elements in 4> cycles of / letters each, 7 cycles of g letters, . . . , where/, g, ... 
are distinct integers > 1, and obtained the result 
fct>\g'y\... \2\ Sr4\ ' ' ' ^^ ^ n\)' 
C. de Polignac*^ investigated at length the highest power of n! dividing 
(nx) \/{x\y. Let rip be the sum of the digits of n to base p. Then 
{x-\-n)p = Xp+np-k{p -1) , {xn)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. Sch6nbaum^° gave a simplified exposition of Landau's first paper.^^ 
S. K. Maitra^i proved that (n - 1) (2n - 1) . . . { (n - 2)n - 1 } is divisible by 
(n — 1) ! if and only if n is a prime. 
E. Stridsberg^^ gave a very elementary proof of Hermite's^^ result. 
E. Landau^^ corrected an error in his^'* proof of the result in No. Ill of his 
paper, no use of which had been made elsewhere. 
Birkeland^^ of Ch. XI noted that a product of 2^k consecutive odd in- 
tegers is^l (mod 2^). 
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, 102. 
B. Pascal, Oeuvres, 3, 1908, 278-282. 
Gioachino Pessuti, Memorie di Mat. Soc. Italiana, 11, 1804, 446. 
W. H. Miller, Jour, fiir Math., 13, 1835, 257. 
S. S. Greatheed, Cambr. Math. Jour., 1, 1839, 102, 112. 
Proofs that multinomial coefficients are integers were given by: 
C. F. Gauss, Disq. Arith., 1801, art. 41. 
Lionnet, Complement des elements d'arith., Paris, 1857, 52. 
V. A. Lebesgue, Nouv. Ann. Math., (2), 1, 1862, 219, 254. 
Factorials Dividing the Product of Differences of r Integers. 
H. W. Segar^" noted that the product of the differences of any r distinct 
integers is divisible by (r — l)!(r — 2)!. . .2!. For the special case of the 
integers 1, 2, . . ., n, r+1, the theorem shows that the product of any n 
consecutive integers is divisible by n!. 
A. Cayley®^ used Segar's theorem to prove that 
m{m — n) . . .{m — r — ln)-rf 
is divisible by r! if m, n are relatively prime [a part of Hermite's-^ result]. 
Segar®^ gave another proof of his theorem. Applying it to the set 
^8aQuar. Jour. Math., 33, 1902, 174-9. "Bull. Soc. Math. France, 32, 1904, 5-43. 
"Casopis, Pras, 34, 1905, 265-300 (Bohemian). 
"Math. Quest. Educat. Times, (2), 12, 1907, 84-5. 
^^Acta Math., 33, 1910, 243. "Nouv. Ann. Math., (4), 13, 1913, 353-5. 
soMessenger Math., 22, 1892-3, 59. "Messenger Math. 22, 1892-3, p. 186. Cf. Hermite." 
^Hbid., 23, 1893-4, 31. Results cited in I'interm^diaire des math., 2, 1895, 132-3, 200; 5, 1898, 
197; 8, 1901, 145. 
