New solvability conditions for congruence ax≡b (mod n)

K. Bibak et al. [arXiv:1503.01806v1 [math.NT],March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x₀ with t = gcd(x₀, n) if and only if gcd thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121-148] and O. Grošek et al. [ibid. 7 (2013), 217-224]. We show that this generalized result for arbitrary t follows from that for t = 1 proved in the later papers. Then we shall analyze this result from the point of view of a weaker condition that gcd . We prove that given integers a, b, n ≥ 1 and t ≥ 1, congruence ax ≡ b (mod n) has a solution x₀ with t dividing gcd(x₀, n) if and only if gcd divides gcd .