214 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
and 
in ={ 77 . 3 + 176 + 56) (77 + 176. 6 + 56) (77 + 176 + 56) 
= 155, 265 (mod 308). 
(21.) The number of numbers which belong to a g’iven exponent when the modulus 
is a power of a prime. 
I. Let the prime be an odd prime, and the modulus. 
In Proposition (12), Corollary, we saw that for mod primitive roots exist. 
Let p be a primitive root. 
The numbers g, g^, . . . (modp^) are congruent to the complete set of 
numbers less than and prime to it. 
The exponent to which any one of these numbers, g\ belongs is t, where 
^ (?/) = Ki 1 
and > and t and cr are co-prime (Prop. 4). 
s = Kcr 
For nny given value of t the value of /c — (^ (p^)A is given, 
cr may then have any value prime to t such that 
•s- > (j) (^C), 
i.e., 
Kcr Kt 
<T t. 
Hence cr may have each of the ^ {t) values of the numbers less than t and prime 
to it. 
Therefore there are <^>{1) numbers having t as their exponent (mod m). 
II. Let the modulus be 2'^. («■ ^ 3). 
In this case we have seen (Proposition 15) that there are 2® numbers with exponent 2* 
(if s > 1): and 3 numbers with exponent 2 ; and 1 number with exponent unity. 
When the modulus is 2^ there is one number with exponent 2 and one with unity. 
When the modulus is 2 there is one number (unity) with exponent unity. 
Deji'iiition. —When a number m is expressed in the form m = 2 ''P/‘P 3 ^^ . . . where 
Pj, Pg, . . . are different odd primes, it will be convenient to speak of 2"', PC- . . . 
as the principal factors of m. 
(22.) The number of numbers, each of which has, as exponent, some power of a 
prime j), for modulus m, p being a divisor of ^ {m). 
Let 
m = 2'‘PC'P2^t • . • 
(j) (PC^) =: . . . 
(/, (PC^) = . . . 
&c. 
