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Let $a$, $b$, $c \in \mathbb{N}$. $[a, b]$ denotes $\mathrm{lcm}(a, b)$ and $(a,b)$ denotes $\gcd(a, b)$

Show that

  1. $(a,[b,c]) = [(a,b),(a,c)]$.
  2. $[a,(b,c)] = ([a,b],[a,c])$.
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    Certainly prime factorization works, but it would be nice to see a proof that avoids it.2011-09-14
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    These properties say that $\mathbb N$ with the divisor relation is a [distributive lattice](http://en.wikipedia.org/wiki/Distributive_lattice).2011-09-14
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    as @Srivatsan Narayanan said, it would be nice to see a proof by definition. I donot know if every such properties that is something (equations) about gcd and lcm hold in ufd always hold in a [GCD](http://en.wikipedia.org/wiki/Gcd_domain) domain?2011-09-14

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