FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
209 
and 
therefore - 
therefore 
Exp of 10 mod 37 = 3, 
37^ divides 10® — 1, 
exp of 10 mod 37® = 3.37" \ 
exp of 10 mod 37". 3®" = 3®"~^. 37"~h 
the number of figures in the period. 
(18.) We will next briefly consider the residues of successive powers of a number 
not prime to the modulus. 
Let a be the number, m the modulus. 
Let m = _pP : where j) consists of powers of those primes which occur as factors 
of a, and P is prime to a. 
Consider the series of residues 
a . . o® . . . (mod m). 
Suppose that the first term which is repeated is rP and suppose that 
a’’(mod m), 
for which we seek the smallest values of r and t. 
Then after the first r — I terms we shall have a period of t terms constantly 
repeated. 
We have 
cP(o* — 1) = 0 (mod Pp), 
and P is prime to a. Therefore 
a* — 1=0 (mod P).(i.). 
Each prime factor of 'p is a factor of «, therefore cd — 1 is prime to p. Therefore 
rd = 0(mod^)).(ii.). 
(i.) Shows that t is the exponent of a for modulus P. 
(ii.) Shows that r is the least number that makes ct'' divisible by p. 
Corollaries .—(i.) When a=l (mod P) ^ = 1 and the period consists of one 
term only, (ii.) When a is divisible by p> the period starts from the first term, 
(iii.) When both these hold good, then every j^ower of a is = a (mod m). 
Examples. —Pesidues of powers of 2 mod 100. 
100 = 53 . 
Exd of 2 mod 5 = 4 
MDCCCXCTIT.—A. 
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