596 
NUMBERS. 
[795 
properly denotes the number of integers not greater than N and prime to it; so that, 
when N = 1, we have 1 an integer not greater than JV and prime to it; but in every 
other case the two definitions agree. 
6. If N, N', are numbers prime to each other, then </> (NN') = cf> (JSf) </> (A"'), and 
so also for any number of numbers having no common divisor; in particular, 
</> (a a №cv ...) = </> (a a ) (b p ) cj) (c y ) ...; <f> (a a ) = a a ^1 — -'j , 
and the theorem is at once verified. We have N = (A'), where the summation 
extends to all the divisors A' of A, unity and the number A itself being included ; thus 
15 = (f) (15) + cf> (5) + <f> (3) + <j> (1), = 8 + 4 + 2 +1. 
7. The prime factor of the binomial function x N —1 is 
_ (cc N — 1) (x Nlab — 1) ... 
^ (x N ' a -l)(x N l b -1) ... ’ 
a rational and integral function of the degree c£(A); say this is called [x N — 1], and 
we have x N — 1 = II [x N ' — 1], where the product extends to all the divisors N' of A, 
unity and the number A included. For instance 
[> 15 -1] = ^ =^-x r + ^-c^ + ^-x+ 1; 
and we have 
x 15 — 1 = \cc 15 — 1] [« 5 — 1] [¿c 3 — 1] \cc — 1], 
= (¿c 8 — x 7 + ... — x + 1)(¿c 4 + x 3 + x i 2 + x + 1) (x* + x + 1) (x — 1). 
8. Congruence to a given modulus. A number x is congruent to 0, to the 
modulus N, x= 0 (mod. N), when x is divisible by N; two numbers x, y are congruent 
to the modulus N, x = y (mod. N), when their difference x — y divides by N, or, what is 
the same thing, if x — y = 0 (mod. N). Observe that, if xy = 0 (mod. N), and x be prime 
to N, then y = 0 (mod. N). 
9. Residues to a given modulus. For a given modulus N we can always find, 
and that in an infinity of ways, a set of N numbers, say N residues, such that every 
number whatever is, to the modulus N, congruent to one and only one of these 
residues. For instance, the residues may be 0, 1, 2, 3, ..., N— 1 (the residue of a 
given number is here simply the positive remainder of the number when divided by 
N); or, N being odd, the system may be 
i 1? ± 2, ..., + \ (N — 1), 
and N even, 
0, +1, ±2,..., ±i(^ — 2), +^A. 
10. Prime residues to a given modulus. Considering only the numbers which are 
prime to a given modulus JST, we have here a set of </> (A) numbers, say <f> (N) prime 
residues, such that every number prime to N is, to the modulus N, congruent to