Step 1 :
Divide $ 522 $ by $ 213 $ and get the remainder

$$ 522 : 213 = 2 \text{ ( remainder is } \color{blue}{ 96 } \text{ ) } $$

The remainder is positive ($ 96 > 0 $), so we will continue with division.

Step 2 :
Divide $ 213 $ by $ \color{blue}{ 96 } $ and get the remainder

$$ 213 : 96 = 2 \text{ ( remainder is } \color{blue}{ 21 } \text{ ) } $$

The remainder is still positive ($ 21 > 0 $), so we will continue with division.

Step 3 :
Divide $ 96 $ by $ \color{blue}{ 21 } $ and get the remainder

$$ 96 : 21 = 4 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

The remainder is still positive ($ 12 > 0 $), so we will continue with division.

Step 4 :
Divide $ 21 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 21 : 12 = 1 \text{ ( remainder is } \color{blue}{ 9 } \text{ ) } $$

The remainder is still positive ($ 9 > 0 $), so we will continue with division.

Step 5 :
Divide $ 12 $ by $ \color{blue}{ 9 } $ and get the remainder

$$ 12 : 9 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

The remainder is still positive ($ 3 > 0 $), so we will continue with division.

Step 6 :
Divide $ 9 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 9 : \color{blue}{ \boxed { 3 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.

This solution can be visualized using a Venn diagram.

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

GCD Calculator