224 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Now we can show that this cannot be unless 
= 1 , = 1 , . . . (mod m). 
For if not suppose that at least m ^ 0 (modulus 
Then 
(mod m), 
(mod ra). 
Let 
I 3 = 4 (mod dg) and the same for I 3 , &c. : and — 4 (mod 
= 0 (mod 
so that < 4 
gi'- = 1 (mod m) 
= A 4 '~(mod m), 
then 
. . . = G 4 ^ (mod m), 
where 
Ii ^ 0 (mod 4 )j 
and 
= 0 (mod 
so that Ig < 4 
I. - 1 
(mod m) 
which is contrary to the supposition that the generators G are independent, and that 
therefore no two numbers of the 4^2 • • • f^at they generate shall be congruent. 
Therefore we must have 
4 = 0 (mod i\), 
= 0 (mod t'o), 
&c. 
and so 
111 ' = 1 (mod ni), 
hd'' = 1 (mod ni), 
and so 
&:c. 
G 4 = ^4 (mod m), 
&c. 
We have shown, then, that when a product of powers of the generators G, 
Gf'Gg'^ ... is congruent to a number with a power of a prime p as exponent, then each 
factor, G]_’S Gg'S . . . must have a power of for its exponent. 
Thus, of the factors of the product to which G^ is congruent, each factor having a 
principal factor of 4 as exponent (Proposition 10),only one factor (viz., that which has, 
as exponent, a power of p) is effective in any powder of G]^ which can be used to form a 
