About formulating $\gcd(x,y,z)=1$

Question

Let $x,y,z$ be three positive integers written as $$x=\prod_{i=1}^{r}p_{i}^{k_{i}}, \quad y=\prod_{i=1}^{r}p_{i}^{l_{i}}, \quad z=\prod_{i=1}^{r}p_{i}^{m_{i}}$$ where $$p_1<p_2<\ldots<p_{r}$$ are prime numbers and $k_{i},l_{i},m_{i}$ are integers. I want to translate the sentence: $$\gcd(x,y,z)=1$$ i.e., $x,y$ and $z$ are coprime into a mathematical formulation by using $p_1,p_2,\ldots,p_{r}$ and $k_{i},l_{i},m_{i}$.

Answer 1

In my previous answer I assumed the numbers were pairwise coprime but if the condition is that there is simply no prime which divides all three then you can express this as $$\sum_{i=1}^r k_il_im_i=0. $$