FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
245 
Thus 
.jfp-'-p'np'-r' _ , , is prime to mjK, 
and, therefore, since a and hjK are also prime to m/K:, the congruence (ii.) is soluble (if 
possible) by means of the tables givmg the indices of numbers prime to the modulus. 
Each solution f of the congruence (ii.) gives a single solution x = ^p^'p'^' . . . 
(mod 7n) of the congruence ax’‘ = b (mod m). 
Example .—Solve 
22a;® = 54 (mod 672). 
The congruence is 
2. 11. a;® = 2. 3^ (mod 2®. 3. 7).(i.) 
Dividing by 2 we have 
11. a;® = 3® (mod 2h 3. 7).(ii.) 
and any solution x of this gives the two solutions a;, a; + 336 of the congruence (i.). 
From (ii.) we get, dividing by 3, 
/ytO 
11 = 9 (mod 2h 7). 
In congruence (ii.) let x — 3f. 
Then 
11. 3h ^® = 9 (mod 2h 7) 
11. 3®. ^®= 1 (mod 2h 7). 
Now (see table, p. 222) 
11 =(4. 3. 0), 
9 = (2. 2. 0), 
therefore 
9. 11 =(0. 1. 0), 
therefore 
f = (0. 3. 0), 
therefore 
Hence (ii.) has the solution 
^=(0. 3. 0) 
^=43 (mod 2\ 7). 
a; = 129 (mod 336), 
and therefore the solutions of (i .) are 
x*= 129, 465 (mod 672). 
