CHAP. II] SYSTEM OF LINEAR CONGRUENCES. 91 



For the latter system, the modulus 5 divides the determinant D. Hence 

 if some minor of order n v is not divisible by 5, while all minors of higher 

 order are divisible by 5, the solution involves exactly v arbitrary parameters 

 and there are 5" sets of solutions. 



L. Kronecker 245 deduced from his theory of modular systems the theorem 

 that, for p a prime, the general solution of 



Z V ik X k = (mod p) (t= 1, ..-,0 



/fc=i 



involves T r independent parameters if the matrix of the tr numbers V<k 

 is of rank r modulo p. 



K. Hensel 246 considered a system of m linear homogeneous congruences 

 hi n unknowns in which the coefficients and the modulus P are either 

 integers or rational integral functions of one variable. We may replace 

 the system by an equivalent system whose modulus divides P and hence 

 finally obtain modulus unity. 



E. Busche 247 proved that the number of solutions of a system of n linear 

 homogeneous congruences in n unknowns equals the modulus if the latter 

 divides the determinant of the system. This theorem is equivalent to the 

 following. Write a ~ b if a b is an integer. If the at,- are integers of 

 determinant =f= 0, the number of non-equivalent solutions x it - , x n of 

 anXi + + ainXn ~ (i = 1, , n) is the absolute value of the deter- 

 minant aij | . 



G. B. Mathews 248 noted that a system of n linear congruences in which 

 the moduli are mi, , m n respectively may be reduced to a system with 

 the same modulus m (the l.c.m. of mi, -, ra n ), by multiplication by m/mi, 

 , m/m n respectively. For the case of a common modulus m the method 

 is to derive an equivalent system of congruences involving respectively 

 n, n 1, -, 1 unknowns. Details are given only for the example 



ax + by + cz = d, a'x + b'y + c'z = d', a"x + b"y + c"z = d" (mod m) . 



Let 8 be the g.c.d. of a, a', a" and let d = pa + qa' + ra" '. Multiplying 

 the congruences by p, q, r respectively and adding, we get a congruence 

 8x + /ft/ + 72 = 5. If, for example, p is prime to m, we get an equivalent 

 system by taking the latter in place of the first congruence of the system. 

 Then eliminate x from the second and third by means of Qx -\- 

 L. Gegenbauer 249 showed that the system of linear congruences 



b k+p y k = (mod p) (p = 0, -, p - 2) 



k=0 



has as many linearly independent sets of solutions as 



p-2 



Z b k x k = (mod p) 



fc=0 



246 Jour, fur Math., 99, 1886, 344; Werke, 3, I, 167. Cf. papers 24-26, p. 226, and 43, p. 232 

 of Vol. I of this History. 



246 Jour, fur Math., 107, 1891, 241. 



247 Mitt. Math. Gesell. Hamburg, 3, 1891, 3-7. 



248 Theory of Numbers, 1892, 13-14. 



249 Monatshefte Math. Phys., 5, 1894, 230. Further report on p. 229 of Vol. I of this History. 



