Step 1 :
Divide $ 1331 $ by $ 1001 $ and get the remainder

$$ 1331 : 1001 = 1 \text{ ( remainder is } \color{blue}{ 330 } \text{ ) } $$

The remainder is positive ($ 330 > 0 $), so we will continue with division.

Step 2 :
Divide $ 1001 $ by $ \color{blue}{ 330 } $ and get the remainder

$$ 1001 : 330 = 3 \text{ ( remainder is } \color{blue}{ 11 } \text{ ) } $$

The remainder is still positive ($ 11 > 0 $), so we will continue with division.

Step 3 :
Divide $ 330 $ by $ \color{blue}{ 11 } $ and get the remainder

$$ 330 : \color{blue}{ \boxed { 11 }} = 30 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 11 }} $.

This solution can be visualized using a Venn diagram.

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

GCD Calculator