Is $\operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c))$

Question

is $\operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c))$? it should be true for positive integers because the commutative algebra by Atiyah says addition, intersection on the ideals of $\mathbb{Z}$ is distributive. how should i solve it?

Answer 1

One method that works in any gcd domain is to eliminate lcms using $\,{\rm lcm}(x,y) = xy/(x,y).\ $ Doing so and clearing denominators, then cancelling $a$ reduces it to

$$(a,b)(b,c)(c,a)\, =\, (a,b,c)(ab,bc,ca)$$

which is true since both sides $= (aab,aac,abb,abc,acc,bbc,bcc)$ by using "gcd polynomial arithmetic" (associate, commutative, distributive laws) to expand the products.

Answer 2

Well its not elegant but let $a=\prod p_i^{a_i} $, $b=\prod p_i^ {b_i} $ and $c=\prod p_i^{ci} $ where $p_i $ are prime and the powers can be zero.

The power of the $p_i $ factor of $lcm (a,\gcd (b,c))$ is $T=\max (a_i,\min (b_i,c_i)) $

The power for $\gcd (lcm (a,b),lcm (a,c))$ is $S=\min (\max (a_i,b_i),\max (a_i,c_i)) $.

If $a_i$ is the least of the three then $T=\min (b_i,c_i) $ and $S=\min (b_i,c_i) $ so equality holds.

If $a_i $ is the most of the three $S=T=a_i $ and equality holds.

If $a_i $ is strictly in between, then $a_i > \min (b_i,c_i) $ so $T=a_i $. Likewise one of $b_i$ or $c_i $ is greater than $a_i $ and the other smaller. So $a_i=$ one of the max pairs, but max of all three is the other max pair. So $S $ is the min of the pairs is $a_i $. So equality holds.