GCD and LCM

Let $a=\prod\limits_{i=1}^{n}p_{i}^{a_{i}}, b=\prod\limits_{i=1}^{n}p_{i}^{b_{i}}, c=\prod\limits_{i=1}^{n}p_{i}^{c_{i}},$ for primes $p_{i}$ .

$\frac{(\gcd(a,b,c))^{2}}{\gcd(a,b)\gcd(b,c)\gcd(c,a)}=\prod\limits_{i=1}^{n}p_{i}^{2\min(a_{i},b_{i},c_{i})-\min(a_{i},b_{i})-\min(b_{i},c_{i})-\min(c_{i},a_{i})}$ $\frac{(lcm(a,b,c))^{2}}{lcm(a,b)lcm(b,c)lcm(c,a)}=\prod\limits_{i=1}^{n}p_{i}^{2\max(a_{i},b_{i},c_{i})-\max(a_{i},b_{i})-\max(b_{i},c_{i})-\max(c_{i},a_{i})}$

WLOG, for some value of $i$ , say $a_{i}\leq b_{i}\leq c_{i}$ . Then, $2\min(a_{i},b_{i},c_{i})-\min(a_{i},b_{i})-\min(b_{i},c_{i})-\min(c_{i},a_{i})=2a_{i}-a_{i}-b_{i}-a_{i}=-b_{i}$ and $2\max(a_{i},b_{i},c_{i})-\max(a_{i},b_{i})-\max(b_{i},c_{i})-\max(c_{i},a_{i})=2c_{i}-b_{i}-c_{i}-c_{i}=-b_{i}$ Therefore, the exponents on each $p_{i}$ are equal in both expressions, as each expression is symmetric in $a_{i},b_{i},c_{i}$ , so both expressions are equal.