how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?
thanks in advance
be L a lattice
L=(N0,LCM,GCD)
in which N0= is da naturals set with zero included
LCM is the Lowest Common Multiple
GCD is the great common divisor
And i want need to prove that lattice L is dristibutive which is
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).
In this case i need to prove,
GCD(a,LCM(b,c)) == LCM(GCD(a,b),GCD(a,c))