GCD and LCM Distribute Over Each Other

Theorem
Let $$a, b, c \in \Z$$.

Then:
 * $$\operatorname{lcm} \left\{{a, \gcd \left\{{b, c}\right\}}\right\} = \gcd \left\{{\operatorname{lcm} \left\{{a, b}\right\}, \operatorname{lcm} \left\{{a, c}\right\}}\right\}$$;
 * $$\gcd \left\{{a, \operatorname{lcm} \left\{{b, c}\right\}}\right\} = \operatorname{lcm} \left\{{\gcd \left\{{a, b}\right\}, \gcd \left\{{a, c}\right\}}\right\}$$.

That is, greatest common divisor and lowest common multiple are distributive over each other.

Proof

 * We show that lowest common multiple is distributive over greatest common divisor.

Let $$p_s$$ be any of the prime divisors of $$a, b$$ or $$c$$, and let $$s_a, s_b$$ and $$s_c$$ be its exponent in each of those numbers.

Let $$x = \operatorname{lcm} \left\{{a, \gcd \left\{{b, c}\right\}}\right\}$$.

Then from GCD and LCM from Prime Decomposition, the exponent of $$p_s$$ in $$x$$ is $$\max \left\{{s_a, \min \left\{{s_b, s_c}\right\}}\right\}$$.

From Max and Min Distributive, $$\max$$ distributes over $$\min$$.

Therefore $$\max \left\{{s_a, \min \left\{{s_b, s_c}\right\}}\right\} = \min \left\{{\max \left\{{s_a, s_b}\right\}, \max \left\{{s_a, s_c}\right\}}\right\}$$ and the result follows.


 * The same argument can be used to show that greatest common divisor is distributive over lowest common multiple, except this time using the result from Max and Min Distributive that $$\min$$ distributes over $$\max$$.