CHAPTER IX. 
DIVISIBILITY OF FACTORIALS AND MULTINOMIAL COEFFICIENTS. 
Highest Power of a Prime Dividing ml 
Genty^ noted that the highest power of 2 dividing (2'*) ! is 2^""^ and the 
quotient is 3"-'(5-7)"-2(9-lM3-15)"-^(17. . .31)""^ . .(2"-l). In general 
if P = 2"'+2"'+. . .+2% where the n's decrease, the highest power of 2 
dividing P! is 2^-^ 
A. M. Legendre^ proved that if p" is the highest power of the prime p 
which divides m !, and if [x] denotes the greatest integer ^ x, 
where s = ao+ • • • +«« is the sum of the digits of m to the base p: 
Th. Bertram^ stated Legendre's result in an equivalent form. 
H. Anton* proved that, U n = vp+a, a<p, v<p, and p is a prime, 
= (p — l)'a!y! (mod p), 
n! 
P 
while, if v = vp-\-a, a'<p, v'<p, 
■^,= {p-iy+^'a\a\v\v\ (modp). 
D. Andr^^ stated that the highest power p" of the prime p dividing n! 
is given expHcitly by }i=^lZi[n/p^] and claimed that merely the method of 
finding ii had been given earHer. He appHed this result to prove that the 
product of n consecutive integers is divisible by n!. 
J. Neuberg*' determined the least integer m such that m\ is divisible by 
a given power of a prime, but overlooked exceptional cases. 
L. Stickelberger^ and K. HenseP gave the formula [cf. Anton*]. 
(2) ^^(-irao!ai!...aj(modp). 
F. de Brun^ wrote g[u] for the exponent of the highest power of the 
prime p dividing u. He gave expressions for 
■^ rP{n;k)=Uf\ g[rP(n;k)] 
3 = 1 
in terms of the functions h{a; k) = l*+2*-f . . . +a^. A special case gives (1). 
^Hist. et M6m. Ac. R. Sc. Inscript. et Belles Lettres de Toulouse, 3, 1788, 97-101 (read Dec. 4, 
1783). 
''Th^orie des nombres, ed. 2, 1808, p. 8; ed. 3, 1830, I, p. 10. 
'Einige Satze aus der Zahlenlehre, Progr. Coin, Berlin, 1849, 18 pp. 
*Archiv Math. Phys., 49, 1869, 298-9. 
"Nouv. Ann. Math., (2), 13, 1874, 185. 
^Mathesis, 7, 1887, 68-69. Cf. A. J. Kempner, Amer. Math, Monthly, 25, 1918, 204-10. 
'Math. Annalen, 37, 1890, 321. 
sArchiv Math. Phys., (3), 2, 1902, 294. 
»Arkiv for Matematik, Astr., Fysik, 5, 1904, No. 25 (French). 
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