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## Main Question or Discussion Point

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR.

I know it can be shown that a AND (b OR c) >= (a AND b) OR (a AND c) for a general lattice, and that if we can show the opposite, that a AND (b OR c) <= (a AND b) OR (a AND c) that implies the two are equal. How do I prove this second part? I am not experienced with number theory, and I have struggled to get a meaningful expression of gcd's and lcm's.

Alternatively, is there a different way you can show me how to prove this?

Thank you!

I know it can be shown that a AND (b OR c) >= (a AND b) OR (a AND c) for a general lattice, and that if we can show the opposite, that a AND (b OR c) <= (a AND b) OR (a AND c) that implies the two are equal. How do I prove this second part? I am not experienced with number theory, and I have struggled to get a meaningful expression of gcd's and lcm's.

Alternatively, is there a different way you can show me how to prove this?

Thank you!