3. From the numerical identities 



(lp + r) (tp + s) = (lt)p + (r s), 



(lp + >*) (tp + s) = (ftp + ls + rt)p + rs, 



we obtain the following formulae for the addition, subtraction and 

 multiplication of classes of residues: 



CrC,= C r , 9 Cr-C\=C rs . 



If two given classes C r and C s , C s =%=Co, lead uniquely to a 

 third class C x such that C r = C a C x , then C x is said to be the quotient 

 of C r by C s and the following notation employed 



The condition for the quotient is evidently identical with the condition 

 that there exist a solution x of the equation 



1) r = sx -f tup- 



In order that a solution x shall exist for r and s arbitrary integers 

 such that s is not divisible by p, it is necessary and sufficient that p 

 be a prime number. To prove the condition necessary , let p=PiP^ 

 where p > 1 , p 2 > 1 . Then 1) can not always be satisfied ; for 

 example, when s = p and r is not divisible by p v The condition 

 that p be a prime is, moreover, a sufficient one by the corollary 

 of 4. Hence the division of classes of residues, the divisor being 

 other than the class (7 , is always possible if, and only if, the modulus p 

 be a prime number. 



In particular, these remarks show that the classes of residues 

 with respect to a prime modulus may be combined by the rational 

 operations of algebra and that each result is itself one of the classes 

 of residues. For example, let p = 3. Then 



G!-}- G 2 = G , Gg-T'Gg 880 GJJ G 2 'G 2 =G 1 



1 , 



4. Format's Theorem. If an integer a be not divisible by a 

 prime number p, then a? 1 1 (mod p). 



Since the integers a, 2 a, 3 a, . . ., (ptya are all distinct 

 modulo jp, their residues must be identical, apart from their order, 

 with the integers 1,2, 3, . . ., p 1. 



Forming the product of the integers in each set, we have 



aP ~ l 1 2 . 3 . . . (p - 1) = 1 2 - 3 . . . (p - 1) (mod p). 



Corollary. - - If a be not divisible by the prime number p, there 

 exists an unique solution of the congruence ax = b (modp). 



Applying the theorem just proven, the solution is evidently 



x aP~ 2 b (mod p). 



