Following the idea of Exercises 6 and 23, outline a constructive method for finding a solution in Z of the congruence ax ≡ b (mod n) for nonzero a, b, n ∈ Z, if the congruence does have a solution. Use this method to find a solution of the congruence 22x ≡ 18 (mod 42).

Data from Exercise 23

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.