FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
211 
Powers of 57 . 5® = 890625 mod 1000000. 
and 
therefore 
and 
Exp of 57 . 5® mod 2 ® is 1 (890625 = 1 (mod 2 ®)), 
57 . 5® is divisible by 5®, 
t = 1 
r = 1, 
and so all powers of 890625 are = 890625 (mod 1000000). 
890625 
890625 
453125 
81250 
3750 
25 
890625 
(19.) Let m = PiP^Ih • • • > where p^, . . . are co-prime. 
Take any number a prime to m. 
Suppose that 
a = (modP i), 
= (modpa), 
= ag (mod 2h), 
&c. 
Since a is prime to on and therefore to it follows that is prime to p,. So is 
prime to pg, to pg, &c. 
Suppose now that, conversely, a^, a..,, ag, . . . prime to P|, pj, pg, . . . are given, and 
we wdsh to find a, such that it is congruent to (modpi), a 2 (modp 2 ), &c. 
Let X| be determined from 
a^iPaPg . . . = 1 (modpi). 
which can be done, and in one way only, since 2hPi • • • is prime to pi; and when 
IS found let a;|Popg . . . = Determine similarly fo, ^3 • ■ • Then the number a is 
given by 
a = + “ 3 ^ + • • • (mod on). 
2 £ 2 
