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

thanks in advance

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- January 12th 2012, 05:36 AMJK64proof lattice
how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?

thanks in advance - January 12th 2012, 06:20 AMPlatoRe: proof lattice
- January 12th 2012, 06:29 AMJK64Re: 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)) - January 12th 2012, 06:31 AMJK64Re: proof lattice
I've already starting thinking about 4 possible cases,

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) - January 12th 2012, 06:38 AMJK64Re: proof lattice
about the iv)

i'm thinking about this:

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?