It's good to have two pupil at home.
They explain me that
gdc(a,b) . lcm(a,b) = a . b
Equally nice to have i-net it explain that because
max(a,b) + min(a,b) = a + b
Can you see connection?
Trivial explanation is in fact that divisibility define lattice on R with meet given by gdc and join by lcm.
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