(a) Prove that the converse of Theorem 1.2 holds in the case of relatively prime integers: For all integers a and b, if there exist integers u and v such that au + bv = 1,

then (a, b) = 1.

(b) Show by counterexample that the converse of Theorem 1.2 does not hold in general. That is, there exist integers a, b, d, u, v, a and b not both 0 and d > 0,

such that d = au + bv but d is not the greatest common divisor of a and b.