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Math Help - proof lattice

  1. #1
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    proof lattice

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

    thanks in advance
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  2. #2
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    Re: proof lattice

    Quote Originally Posted by JK64 View Post
    how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?
    What does that mean?
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  3. #3
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    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))
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  4. #4
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    Re: 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)
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  5. #5
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    Re: 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?
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