Step 1 :
Divide $ 564 $ by $ 234 $ and get the remainder

$$ 564 : 234 = 2 \text{ ( remainder is } \color{blue}{ 96 } \text{ ) } $$

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

Step 2 :
Divide $ 234 $ by $ \color{blue}{ 96 } $ and get the remainder

$$ 234 : 96 = 2 \text{ ( remainder is } \color{blue}{ 42 } \text{ ) } $$

The remainder is still positive ($ 42 > 0 $), so we will continue with division.

Step 3 :
Divide $ 96 $ by $ \color{blue}{ 42 } $ and get the remainder

$$ 96 : 42 = 2 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

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

Step 4 :
Divide $ 42 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 42 : 12 = 3 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

The remainder is still positive ($ 6 > 0 $), so we will continue with division.

Step 5 :
Divide $ 12 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 12 : \color{blue}{ \boxed { 6 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 6 }} $.

This solution can be visualized using a Venn diagram.

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

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