294 
MR. G. T BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
therefore 
therefoi'e 
therefore, if 
So 
Example 6. 
and therefore 
(2 + t)^4 = (10. 0), 
= 2 ), 
^={a. h), 
5a = 1 (mod 12), a = 5, 
5h = 2 (mod 4), h = 2, 
^={ 5 . 2 ), 
= 58 (mod 8 + f)- 
= 58 (2 + i) (mod 15 + lOt) 
= 1 + Si. 
2x~ = 26 + 32i (mod 40), 
= 13 + 16f (mod 20). 
Each solution of the latter gives four of the former, viz., 
X, X 20, X + 20g a; + 20 + 20f (mod 40). 
The congruence is 
a;2 = - f (1 4- 2if (4 + i) [mod (1 + if. (2 + i). (1 + 2^], 
therefore 
Let 
Then 
and therefore 
^ ^ ~ + 20 (4 4- 0 (1 + f)h (2 + i)]. 
a: = (1 4- 20 ^(mod 20). 
(1 + 2i) ^^ = — f (1 + 2i) (4 + i) [mod (1 + 0^- (2 4- OJ’ 
f" = — i .{A i) (mod 8 4* ^O^ 
= 1 — 4f (mod 8 + Ai), 
P = 9 (mod 8 + 4^). 
From the tables 
and therefore 
(•i) (i) 
9 = (2. 0. 0), 
e = {'2. 0 . 0 ). 
