CHAP, ii] SYSTEM OF LINEAR CONGRUENCES. 89 



possible if the g.c.d. 6 of b and &' is not a divisor of a'm c f ; but, if 6 be 

 such a divisor, the general solution is of the form z = n + z'b'/6. Thus 

 x = m r + B'z, where B' is the l.c.m. of b, b'. Similarly, also the third 

 fraction will be integral if x = M + Bz, where B is the l.c.m. of b, b', b". 



* M. Fekete 238 treated the general system of linear congruences in one 

 unknown. 



C. F. Gauss 239 discussed at length the solution of n linear congruences 

 in n unknowns. His second (more typical) example is 



3x + 5y + z = 4, 2x + 3y + 2z = 7, 5z + y + 82 = 6 (mod 12). 



We first seek integers 240 , ', " without a common factor such that the sum 

 of their products by the coefficients of y (and by those of z) is congruent 

 to zero: 



5 + 3' + I" = 0, f + 2$' + 3" s (mod 12). 



Thus = 1, ' = 2, " = 1. Multiplying the congruences by these, 

 and adding, we get 4x = 4 (mod 12). Similarly, the multipliers 1, 

 1, 1 give 7y = 5, while the multipliers - 13, 22, 1 give 28z = 96. 

 Thus x = 2 + 3Z, y = 11 (or 11 + 12r), = 3w. The proposed congru- 

 ences now give three equivalent to 



19 + 3 + u = 0, 10 + 2t + 2w = 0, 5 + 5 + 3u = (mod 4), 

 which are all satisfied if and only if u = t + 1 (mod 4). Thus 



(x, y, z*) = (2, 11, 3), (5, 11, 6), (8, 11, 9), (11, 11, 0) (mod 12). 

 H. J. S. Smith 241 noted that the theory was left imperfect by Gauss. In 

 (1) AU.XI + + A in x n = A in+l (mod M} (i = 1, -, n), 



denote the determinant | AH by D. If D is prime to M, there is one 

 and but one set of solutions. Next, let D be not prime to M = pTpT- , 

 where the p's are distinct primes. A necessary condition for solvability is 

 that there be solutions for each modulus p\ Conversely, if there be P t - 

 sets of solutions for modulus p*, there are PiPz sets of solutions modulo 

 M. Hence consider (1) for the modulus p m , and let I r be the exponent of 

 the highest power of p dividing all the r-rowed minors of D. Then, if 

 / I n -i = m, the congruences, if solvable, have p ln sets of solutions. 

 But if I n / n _i > m, we can assign a value of r such that 



Ir+i ~ I r > m^Ir Ir-l 



and then the number (if any) of sets of solutions is p k , where 



k = I r + (n r)m. 



238 Math. 4s Phys. Lapok, Budapest, 17, 1908, 328-49. 



239 Disq. Arith., Art. 37; Werke, I, 27-30. 



240 F. J. Studnicka, Sitzungsberichte, Akad. Wiss., Prag, 1875, 114, noted that they are pro- 



portional to the signed minors of the coefficients of the first column in the determinant 

 of the coefficients. 



241 Report British Assoc. for 1859, 228-67; Coll. Math. Papers, I, 43-5. 



