Prove or disprove the following statements.
(a) If a; b 2 Z, a; b > 0, and a2 j b3, then a j b.
(b) If a; b 2 Z, a; b > 0, and a2 j b2, then a j b.
Find all pairs of positive integers a and b for which (a; b) = 18 and [a; b] = 540. Prove that you've found all such integers.
(a) Let a; b 2 Z. Prove that if a and b are both expressible in the form 6n + 1 with n 2 Z, then so is ab.
(b) Prove that there are in nitely many prime numbers of the form 6n+5 with n 2 Z.