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.

LCM Distributive over GCD
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:
 * $\max \left\{{s_a, \min \left\{{s_b, s_c}\right\}}\right\} = \min \left\{{\max \left\{{s_a, s_b}\right\}, \max \left\{{s_a, s_c}\right\}}\right\}$

Hence it follows that LCM is distributive over GCD.

GCD Distributive over LCM
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 $x = \gcd \left\{{a, \operatorname{lcm} \left\{{b, c}\right\}}\right\}$.

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

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

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

Hence it follows that GCD is distributive over LCM.