244 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
(33.) The last proposition shows how, by means of tables of indices for composite 
moduli, to solve (when possible) the congruence 
% 
ax’‘=h (mod on), 
where m is any composite modulus and a and h are both prime to on. 
To complete the question of tlie solution of the congruence for all cases we have 
now to show how it is solved when a and h are not both prime to on. 
I. First, if a is not prime to on, the congruence can be at once reduced to the case 
in which it is so ; for if a and toi have G.C.M. k, then k must divide h, and we have 
/a\ h , m\ , « . . on 
( - ] x’^ = - { mod — , where now - is prime to — • 
\kJ k \ k J k k 
Any one solution x of this gives k 
(mod on), viz,, 
X, X -, X 
K 
solutions of the original congruence ax'‘=h 
'2on 
K 
, X K — 1 
m 
K 
II. We have now, therefore, only to deal with the congruence of the form 
ax^‘ = h (mod on), in which a is prime to on and b is not prime to ooi. Let k be the 
G.C.M. of h and ooi, and express k as the product of its principal factors k = • • • 
Since /< divides b and on, it divides ax'\ 
Now a is prime to on, and therefore to k; therefore k divides a;", i.e., . . . 
divides x^\ Therefore x is divisible by • • • where 
oip ; o\ 
np = r', 
&c. 
Take p to be equal to o'/oi if o'/oi is an integer. If o'jn is not an integer take p equal 
to the integer next greater. 
X = . . ., 
then the congruence 
h f ^ o)i\ 
a — = - { mod q . 
. . . . (i.) 
K K \ K j 
gives 
u - [mod. 
K \. ^ } 
. . . (ii). 
If np — O' Q (i.e., if o'jn is not an integer), then the coefficient of in this con¬ 
gruence is divisible by a power of and therefore, if the congruence is possible, oii’k 
is not (for then h 'jK would be also, whereas and !)!< are co-prime). 
