Lowest Common Multiple of multiple pairwise coprime integers?

Question

I've seen that the equation $$\text{gcd}(a,b,c)\times \text{lcm}(a,b,c)=abc$$ Is true provided that $a,b,c$ are pairwise coprime. However surely then this equation has no significance as $$\text{gcd}(a,b,c)=\text{gcd}(a,\text{gcd}(b,c))=\text{gcd}(a,1)=1$$ So the statement is just that $$\text{lcm}(a,b,c)=abc$$ for pairwise coprime $a,b,c$. Is my reasoning correct? I feel it would then be simpler to state that for $a_1,a_2,...,a_n$ pairwise coprime we have $$\text{gcd}(a_1,a_2,...,a_n)=1\;,\text{lcm}(a_1,a_2,...,a_n)=a_1a_2\cdot\cdot\cdot a_n$$

Answer 1

You're correct. If fact, there isn't even necessary for them to all be pairwise coprime, it's true if at least one pair of them is coprime.