Let R be a GCD domain.
Let A be an m×n matrix and B an m× 1 matrix with entries in R. Let c≠0,d ∈ R.
We consider the linear GCD equation GCD(AX + B,c) = d. Let S denote its set of
solutions. We prove necessary and sufficient conditions that S be nonempty. An
element t in R is called a solution modulus if X + tRn⊆ S whenever X ∈ S. We show
that if c∕d is a product of prime elements of R, then the ideal of solution moduli is a
principal ideal of R and its generator t0 is determined. When R∕t0R is a finite ring,
we derive an explicit formula for the number of distinct solutions (modt0) of
GCD(AX + B,c) = d.
