21-2 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
For 
and 
Tl'ierefore 
and similarly 
and therefore 
t 
^ 2 , ^ 3 , • • . fii’e ^11 congruent to zero mod_^j^, 
is congruent to unity modp^ 
a = (mod 
= a2 (modp^)^ 
&c., 
a = + a2^3 + “3^ + • • • (mod m). 
a formula wliicli gives the value of a corresponding to given values of the a’s, the 
coefficients ^ being independent of the a’s and depending only on the moduli 
Pn Vi - - 
Note .—The principal use to be made of this proposition and the next will be for 
the special case when p.,, . . . are the powers of primes of which the modulus m is 
the product. 
(20.) Let us take two numljers expressed in tins form, 
in which 
a = a^f^ + a 2 ^o + . . . (mod m), 
h = + • • • (mod m), 
= 1 (mod_^9j) 
■»("»■'3 
with similar relations for ^ 3 , . . . 
Let us form the product of a and 6. 
By Proposition (L8), Corollary (hi.), all powers of are congruent to mod m, 
therefore 
(mod 7/i). 
Taking any cross-term such as since 
= 0 (inodp^p-j . . . ), 
^2 = 0 (modp^7;3 . . . ), 
therefore 
= 0 (mod ill). 
ah = d- + “ 3 ^ 3^3 + • • • ('^md ill), 
Hence 
