242 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Then we must have 
+ nlj = i\ (mod fj) 
G + = i'z (mod t. 2 ), 
&c. 
The G.C.M. of II and must divide i\ — &c., in order that the congruences may 
be soluble. 
These being so we find Ij, lo, . . . by reference to tables with moduli b) • • • which 
are (or may be) composite numbers. 
If 
is the G.C.M. of n and wall have values mod 
V .2 5 ) j) 'ii^ and ^ 0 ) lo 5 ) )) G 5 
kc. 
Hence, by reference again to the table for modulus in, we obtain at once the 
v^, v. 2 > • numbers which satisfy the congruence ax" = h (mod m). 
Let us now solve the same congruence with the help only of tables of indices for 
powers of primes as moduli. 
If 
a^J is a solution of the congruence cfx" = h (mod 2 "), 
otj ,, „ „ ax" = h (mod 
then 
&c., 
X = m) 
is a solution of the congruence 
ax’‘ = h (mod m). 
Hence the solution is conducted as follows :— 
By means of the tables of indices find 
all the values a,) which satisfy ax" = b (mod 2 ''), 
,, ,, „ (mod 
&c. 
Next find the values of the f’s, Proposition (19), each being found b}^ the tables. 
Lastly the values of x which satisfy ax" = h (mod m) must each be calculated sepa¬ 
rately by giving to each of olq, otj, . . . one of its possible values. 
In the case of a multiple solution the labour involved in this last step may be very 
