A=set containing prime factors of a where repitition is allowed in a set is assumed

B=set containing prime factors of b where repitition is allowed in a set is assumed

m=product of the elements of (A union B)

d=product of the elements of (A intersection B)

dm=product of the elements of (A union B) and (A intersection B)

=Product of the elements of A-B ,B-A,(A intersection B),(A intersection B)

=product of the elements of A and B

=ab(proved)