Relation between HCF, LCM and product of multiple numbers

Question

It is well known that for two numbers $a $ and $b$, $$\text {lcm} (a,b)\times \text {hcf} (a,b)=ab$$ Does there exist a similar equality/ inequality between HCF, LCM and product of multiple numbers? (i.e. some relation between $\text {hcf} (a_1,a_2,...,a_n)$, $\text{lcm} (a_1,a_2,...,a_n)$ and $a_1\times a_2 \times ...\times a_n$)

Answer 1

You cannot express $a_1a_2\dots a_n$ in terms of $\gcd(a_1,a_2,\dots,a_n)$ and $\operatorname{lcm}(a_1,a_2,\dots,a_n)$ in any way if $n>2$. For instance, if $n=3$, note that $\{2,4,16\}$ and $\{2,8,16\}$ have the same GCD and LCM, but they have different products. You can find similar examples for any $n>2$.

For an inequality, I don't know what sort of thing you're looking for. But it is straightforward to see that $$\gcd(a_1,a_2,\dots,a_n)\operatorname{lcm}(a_1,a_2,\dots,a_n)\leq a_1a_2\dots a_n$$ whenever $n\geq 2$ (just compare the $p$-adic valuations of both sides for each prime $p$--the left-hand side is the sum of the valuations of two of the $a_i$, while the right-hand side is the sum of the valuations of all of them).

Answer 2

Not with just product.

The product of the lcm and hcf of $(2,2,2)$ is $4$. For $(1,1,8)$ it is $8$. But $2\times 2\times 2=1\times 1\times 8$.

Thus the product of lcm and hcf is not a function of the product of the three numbers.

Similarly, note that $(2,2,4)$ and $(2,4,4)$ have the same lcm and hcf. But their products are different, so the product of three numbers is not a function of their lcm and hcf.