The GCD of given numbers is 31.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}1023 =& 3\cdot11\cdot31\\[8pt]217 =& 7\cdot31\\[8pt]\end{aligned}$$(view steps on how to factor 1023 and 217. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}1023 =& 3\cdot11\cdot\color{blue}{\boxed{31}}\\[8pt]217 =& 7\cdot\color{blue}{\boxed{31}}\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 31 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.