Generalize Exercise 22 by showing that for nonzero a, b, n ∈ Z, the congruence ax ≡ b (mod n) has a solution in Z if and only if the positive gcd of a and n in Z divides b. Interpret this result in the ring Z_n.

Data from Exercise 22

Using the last statement in Theorem 46.9, show that for nonzero a, b, n ∈ Z, the congruence ax ≡ b (mod n) has a solution in Z if a and n are relatively prime.