1. ## proof lattice

how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?

2. ## Re: proof lattice

Originally Posted by JK64
how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?
What does that mean?

3. ## Re: proof lattice

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))

4. ## Re: proof lattice

i) a|/b & a|/c [which is easy to prove]
ii) a|b & a|/c
iii) a|/b & a|c
iv) a|b & a|c

ii) & iii) i think i'm close to the answer, but i'm really struggling with iv)

5. ## Re: proof lattice

If a|b & a|c then

(i) GCD(a,LCM(b,c))=a

on other hand

(ii) LCM( GCD(a,b) , GCD(a,c) )=LCM(a,a)=a

hence, (i)==(ii), what do you think?