Step 1 : Find prime factorization of each number.

$$\begin{aligned}1089 =& 3\cdot3\cdot11\cdot11\\[8pt]666 =& 2\cdot3\cdot3\cdot37\\[8pt]\end{aligned}$$

(view steps on how to factor 1089 and 666. )

Step 2 : Put a box around factors that are common for all numbers:

$$\begin{aligned}1089 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{3}}\cdot11\cdot11\\[8pt]666 =& 2\cdot\color{blue}{\boxed{3}}\cdot\color{red}{\boxed{3}}\cdot37\\[8pt]\end{aligned}$$

Step 3 : Multiply the boxed numbers together:

$$ GCD = 3\cdot3 = 9 $$

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

GCD Calculator