# The elegant expression in terms of gcd and lcm - algebra

Given three positive integer numbers $k_1$, $k_2$, $k_3$, we may denote their greatest common divisor(gcd) by

$\gcd(k_i,k_j)\equiv k_{ij}$ for gcd of a two pair of number $k_i,k_j$.

$\gcd(k_1,k_2,k_3)\equiv k_{123}$ for gcd of all numbers $k_1$, $k_2$, $k_3$.

We also denote least common multiplier (lcm)

lcm$(k_i,k_j)\equiv K_{ij}$ for lcm of a two pair of number $k_i,k_j$.

lcm$(k_1,k_2,k_3)\equiv K_{123}$ for lcm of all numbers $k_1$, $k_2$, $k_3$.

Q1: Can one express $$\gcd(k_1 k_2,k_2 k_3,k_3 k_1)$$ in terms of $k_i$, $k_{ij}$, $k_{123}$, $K_{ij}$, $K_{123}$? (here with $i,j \in \{1,2,3\}$).

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Q2: Can one express $$\gcd(k_1 k_2,k_2 k_3,k_3 k_1)$$ in terms of $k_i$, $k_{ij}$, $k_{123}$? without using $K_{ij}$, $K_{123}$ (here with $i,j \in \{1,2,3\}$).

See a related question.

$\gcd(k_1 k_2,k_2 k_3,k_3 k_1) = \gcd(\gcd(k_1 k_2,k_2 k_3),k_3 k_1) = \gcd(k_2 \gcd(k_1,k_3),k_3 k_1) = \gcd(k_2 \gcd(k_1,k_3),\gcd(k_3, k_1) lcm(k_3, k_1)) = \gcd(k_1,k_3)\gcd(k_2 ,lcm(k_3, k_1)) = k_{13}\gcd(k_2, K_{13})$