New Solvability Conditions for Congruence $ax\equiv b\pmod n$

Abstrakt

K.Bibak et al. [arXiv:1503.01806v1 [math.NT], March 5 2015] proved that congruence ax=b (mod n) has a solution x with gcd(x,n)=t if and only if gcd(a,n/t)=gcd(b/t,n/t) 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(a,n/t) only divides gcd(b/t,n/t). We prove that given integers n>=1,a,b and t>=1, congruence ax=b (mod n) has a solution x with t dividing gcd(x,n) if and only if gcd(a,n/t) divides gcd(b/t,n/t).