248 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Then 
X = — y^T^) 
Y = ij d -{■ arj). 
The second equation gives 2 / = Y (mod d), which determines y. 
Then 
+ OtT? = , 
a and /3 are co-prime, and therefore we can find a' and /3' so that a ^ 
and then 
Therefore, 
and therefore 
^ = a' -f \a 
d 
> where X is some integer. 
V = — ^ 
X = a; + c? 
d 
- X/3 
[aa -j- ^y8 ) X (a^ + yS~) 
— a? -}- (Y — y) [aa -|- ^/3) + X . (od + /^~), 
X 
’ = X — (Y — y) (aa' -j- ^/3') (mod d . a^ -j- 
a, 
which determines x. 
So X and y are given by 
y = Y (mod d) 
cc = X — (Y — y) (aa' -j- /3y8') (mod d . a^ -f /3') 
where aa' -f- yS/3' is a constant depending on the modulus. 
Example. 
Mod 3 (3 + 2^) 
d = 3. 
aa' -f- = — 5. 
The reducing formulae are 
y = Y (mod 3) 
a; = X + 5 (Y — 2/) (mod 39). 
Thus to find the residues of the successive powers of 1 'ii •— 
