Let Z denote the domain of ordinary integers and let m(≧1), n(≧1), li(i=1,…, m; i=1,…, n)∈Z.We consider the solutions x∈Zn of G.C.D. (l11x1 +…+l1nxn+l1,…, lm1x1+…+lmnxn+lm, c)=d, where c(≠0), d(≧1)∈Z and G.C.D. denotes “greatest common divisor”. Necessary and sufficient conditions for solvability are proved. An integer t is called a solution modulus if whenever x is a solution of (1), x + ty is also a solution of (1) for all y ∈ Zn The positive generator of the ideal in Z of all such solution moduli is called the minimum modulus of (1). This minimum modulus is calculated and the number of solutions modulo it is derived.

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Journal Pacific Journal of Mathematics
Citation
Jacobson, D. (David), & Williams, K.S. (1971). On the solution of linear G.C.D. equations. Pacific Journal of Mathematics, 39(1), 187–206.