Why do 42 and 30 have the same structure in the diagramm of Hasse?

Question

I have an exam coming up tomorrow and there's just one more question more to prepare. I would be so gratefull if anyone could help. It´s about the relations of dividers. By the help of the diagram of Hasse: why do 42 and 30 have the same structure?
Can anyone please show me how to solve those kind of tasks? I have sadly no idea.
Thank you! Sophia

Answer 1

$42=2\cdot3\cdot7$ while $30=2\cdot3\cdot5$ since the Hasse diagram ignores what the actual primes are these numbers have the same Hasse diagram. That is, the Hasse diagram only cares if the primes inside the product are different. Every product of exactly three distinct primes, each appearing once has the same diagram as $42$ and $30$, for example $3\cdot5\cdot7=105$ also has that diagram.

Answer 2

More generally, the Hasse diagram of the positive divisors of two positive numbers $m, n$ will be the same (more precisely, (order) isomorphic as posets) if and only if there are distinct primes $p_1, \dots, p_n$ and distinct primes $q_1, \dots , q_n$ such that $$ m = p_1^{e_1} \cdots p_n^{e_n}, \qquad n2 = q_1^{e_1} \cdots q_n^{e_n}. $$