If [tex]A[/tex] is the set of positive divisors of [tex]m[/tex] and [tex]B[/tex] the set of positive divisors of [tex]n[/tex], then [tex]A\cup B[/tex] is the set of positive divisors of [tex]mn[/tex].
Use the inclusion/exclusion principle:
[tex]|A \cup B| = |A| + |B| - |A \cap B|[/tex]
where [tex]|\cdot|[/tex] denotes set cardinality (the number of elements the set contains). The set [tex]A\cap B[/tex] is the set of common divisors of [tex]m[/tex] and [tex]n[/tex]. Then
[tex]14 = 5 + 6 - |A\cap B| \implies |A\cap B| = 3[/tex]
so that [tex]m[/tex] and [tex]n[/tex] share 3 divisors [tex]d_1,d_2,d_3[/tex]; let [tex]k=d_1d_2d_3[/tex] be their product. They must be prime
This means we can write
[tex]m = p_1 p_2 k[/tex]
[tex]n = p_3 p_4 p_5 k[/tex]
[tex]\implies mn = p_1 p_2 p_3 p_4 p_5 k^2 = p_1 p_2 p_3 p_4 p_5 {d_1}^2 {d_2}^2 {d_3}^2[/tex]
so that [tex]mn[/tex] has up to 8 distinct prime factors.
