90 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



Smith 207 wrote V n for the determinant AH \ of (1), V n -i for the g.c.d. 

 of its first minors, , Vi for the g.c.d. of the elements A^, and set Vo = 1. 

 Let D n , D n -i, --, DQ be the corresponding g.c.d. 's for the augmented 

 matrix. Let 5; and di denote respectively the g.c.d. of M, Vv/V;_i, and 

 M, DilDi^i. Set d = di'dn, 8 = 8j_---8 n . Then the system of con- 

 gruences (1) is solvable if and only if d = 8; when this condition is satisfied 

 the number of incongruent sets of solutions is d. There are similar theorems 

 (pp. 402-4) when the number of unknowns is either less or greater than 

 the number of congruences. 



Smith 242 employed a prime factor p of M, and the exponents p., a s , <x s 

 of the highest powers of p dividing M, D s , v, respectively. He proved 

 that his preceding theorems can be replaced by the following: For the 

 modulus p 11 , the congruences (1) are solvable if and only if a a = a ff , where 

 a a a ff _i is the first term of a n a n _i, a n -\ a n _ 2 , which is < /*; 

 when this condition is satisfied the number of incongruent sets of solutions 

 is p k , where k a a + (n <T)H. 



G. Frobenius 210 (pp. 185-194) proved that the congruences (1) have 

 M n ~ l incongruent sets of solutions if the Z-rowed determinants of A have 

 with M no common divisor and if the (Z + 1) -rowed determinants of the 

 augmented matrix of all coefficients are divisible by M, where Z is the rank 

 of the matrix A of the coefficients of the unknowns. If the rank of the 

 augmented matrix is Z + 1 and the g.c.d. of the (Z -f- l)-rowed determinants 

 is d', while the rank of A is Z and the g.c.d. of the Z-rowed determinants is 

 d, the congruences (1) have no solutions if the modulus M is not a divisor 

 of d'jd. The number of incongruent sets of solutions of the homogeneous 

 congruences AnXi -f- + A in x n = (mod M) equals SiS 2 - -s n , where s x 

 is the g.c.d. of M and the Xth elementary divisor of the matrix (An). 



Frobenius 212 proved that a system of linear homogeneous congruences 

 modulo M inn unknowns has a fundamental system of n s sets of solu- 

 tions, but none of fewer than n s, if the determinants of order s + 1 have 

 with M a common divisor but the determinants of order s do not. He in- 

 vestigated the rank and equivalence of linear forms modulo M . 



F. Jorcke 243 treated systems of linear congruences without novelty. 



D. de Gyergy6szentmiklos 244 considered the congruences 



n 



X dpjXj = u p (mod m) (p = 1, , n). 



Let D = | dpj , and V k be the determinant derived from D by putting the 

 it's in the kih column. Let 5 be the g.c.d. of m and D. If any Vk is 

 not divisible by 8, there is no solution. Next, let each V k be divisible by 5. 

 Then Dx k = Vk (mod m) uniquely determines x k = a k modulo m/5. Set 

 %k = otk + thm/8 in the initial congruences. Thus 



+ + a pn t n = w p (mod 5). 



242 Proc. London Math. Soc., 4, 1871-3, 241-9; Coll. Math. Papers, II, 71-80. 



243 Ueber Zahlenkongruenzen und einige Anwendungen derselben, Progr. Fraustadt, 1878. 



244 Comptes Rendus Paris, 88, 1879, 1311. 



