Step 1 :
Divide $ 2121 $ by $ 567 $ and get the remainder

$$ 2121 : 567 = 3 \text{ ( remainder is } \color{blue}{ 420 } \text{ ) } $$

The remainder is positive ($ 420 > 0 $), so we will continue with division.

Step 2 :
Divide $ 567 $ by $ \color{blue}{ 420 } $ and get the remainder

$$ 567 : 420 = 1 \text{ ( remainder is } \color{blue}{ 147 } \text{ ) } $$

The remainder is still positive ($ 147 > 0 $), so we will continue with division.

Step 3 :
Divide $ 420 $ by $ \color{blue}{ 147 } $ and get the remainder

$$ 420 : 147 = 2 \text{ ( remainder is } \color{blue}{ 126 } \text{ ) } $$

The remainder is still positive ($ 126 > 0 $), so we will continue with division.

Step 4 :
Divide $ 147 $ by $ \color{blue}{ 126 } $ and get the remainder

$$ 147 : 126 = 1 \text{ ( remainder is } \color{blue}{ 21 } \text{ ) } $$

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

Step 5 :
Divide $ 126 $ by $ \color{blue}{ 21 } $ and get the remainder

$$ 126 : \color{blue}{ \boxed { 21 }} = 6 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 21 }} $.

This solution can be visualized using a Venn diagram.

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

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