Systems of linear congruences with individual moduli Original Research Article

Consider an n×n matrix A, with integer elements, a column vector x of n integer indeterminates, and a column vector Q of n integers greater than unity. Ax modulo Q constitutes another n-vector b of nonnegative integers. The elemental feature of interest for such systems is whether they are regular (i.e., nonsingular): whether b uniquely determines x modulo Q. Let Pσ denote the permutation matrix corresponding to a permutation σ of {1,2,…,n}. Then, for the special case of all pairs of elements of Q having the same greatest common factor, it is established that regularity obtains if and only if there exists a permutation σ so that PσAPσT is a triangular matrix with each element on the main diagonal coprime to its respective modulus (from PσQ). To resolve systems with general Q, a set of moduli is first derived from each original modulus by factoring it into prime-power factors. We introduce a corresponding regularity-preserving transformation of A and Q into an A′ and Q′: the latter containing, exclusively, prime-power moduli. Elementary transformations of A′ preserving regularity modulo Q′—denoted equivalences—are introduced. A′ is shown to be regular modulo Q′ if and only if there exists a permutation σ so that PσA′PσT is equivalent to a triangular matrix, having each element on the main diagonal coprime to its respective modulus (from PσQ′). Whence, regularity is fully resolved for general systems. An algorithm for solving an arbitrary regular system Ax≡b (mod Q) is, furthermore, implicit in these results.