270 History of the Theory of Numbers. [Chap, ix 
a, a-\-N,. . ., a+A^", we conclude that the product of their differences is 
divisible by n!(n — 1)!. . .21 = p. But the product equals 
p=iN-ir-' (ir--ir-\ . .{N^-'^-iyiN"-'-!), 
multiplied by a power of A^. Hence, if N is prime to n!, P is divisible by v; 
in any case a least number X is found such that N^P is divisible by ;'. It is 
shown that the product of the differences of mi,. . ., m^ is divisible by 
k\{k — l)\. . .2! if there be any integer p such that Wi+p, . . ., nik+p are 
relatively prime to each of 1, 2, . . . , A;. It is proved that the product of any 
n distinct integers multiplied by the product of all their differences is a 
multiple of n!(n-l)!. . .2!. 
E. de Jonquieres^^ and F. J. Studnicka^ proved the last theorem. 
E, B. Elliott^^ proved Segar's theorem in the form: The product of the 
differences of n distinct numbers is di\'isible by the product of the differences 
of 0, 1,..., n — 1. He added the new theorems: The product of the 
differences of n distinct squares is divisible by the product of the differences 
of 0", 1",..., (n — 1)"; that for the squares of n distinct odd numbers, 
multiplied by the product of the n numbers, is divisible by the product of 
the differences of the squares of the first n odd numbers, multiplied by their 
product. 
Residues of Multinomial Coefficients. 
Leibniz^' '^ of Ch. Ill noted that the coefficients in (ZaY—Za^ are 
di\'isible by p. 
Ch. Babbage^^ proved that, if n is a prime, (^n-/) — 1 is divisible by 
n^, while ("p") — 1 is divisible by p if and only if p is a prime. 
G. Libri^'' noted that, if m = 6p-hl is a prime, 
2^p-^-^ep-l-(^^P~^y+(^P~^y'- ..=0 (mod m). 
E. Kummer^^ determined the highest power p^ of a prime p dividing 
^; ^1 > A = ao+aip-{-. . .-{-aip\ B=ho+hip-{- . . .+bip\ 
where the a, and 6, belong to the set 0, 1,. . ., p — 1. We may determine 
Ci in this set and e, = or 1 such that 
(3) ao+6o = €oP+Co, €o+ai+6i = eip+Ci, ei +a2 + ?>2 = €2^4-^2, •■ •• 
Multiply the first equation by 1, the second by p, the third by p^, etc., and 
add. Thus 
A+B = Co-\-Cip+ . . .+Cip'+e,p'+\ 
"Comptes Rendus Paris, 120, 1895. 408-10. 534-7. 
"Vpstnik Ceske Ak., 7, 1898, No. 3, 165 (Bohemian). 
«Messinger Math., 27, 1897-8, 12-15. 
"Edinburgh Phil. Jour., 1, 1819, 46. 
"Jour, fur Math., 9, 1832, 73. Proofs by Stern, 12, 1834, 288. 
"/6ui., 44, 1852, 115-6. Cayley, Math. Quest. Educ. Times, 10, 1868, 88-9. 
