The GCD of given numbers is 19.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}589 =& 19\cdot31\\[8pt]817 =& 19\cdot43\\[8pt]\end{aligned}$$(view steps on how to factor 589 and 817. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}589 =& \color{blue}{\boxed{19}}\cdot31\\[8pt]817 =& \color{blue}{\boxed{19}}\cdot43\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 19 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.