Step 1 :
Divide $ 567 $ by $ 411 $ and get the remainder

$$ 567 : 411 = 1 \text{ ( remainder is } \color{blue}{ 156 } \text{ ) } $$

The remainder is positive ($ 156 > 0 $), so we will continue with division.

Step 2 :
Divide $ 411 $ by $ \color{blue}{ 156 } $ and get the remainder

$$ 411 : 156 = 2 \text{ ( remainder is } \color{blue}{ 99 } \text{ ) } $$

The remainder is still positive ($ 99 > 0 $), so we will continue with division.

Step 3 :
Divide $ 156 $ by $ \color{blue}{ 99 } $ and get the remainder

$$ 156 : 99 = 1 \text{ ( remainder is } \color{blue}{ 57 } \text{ ) } $$

The remainder is still positive ($ 57 > 0 $), so we will continue with division.

Step 4 :
Divide $ 99 $ by $ \color{blue}{ 57 } $ and get the remainder

$$ 99 : 57 = 1 \text{ ( remainder is } \color{blue}{ 42 } \text{ ) } $$

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

Step 5 :
Divide $ 57 $ by $ \color{blue}{ 42 } $ and get the remainder

$$ 57 : 42 = 1 \text{ ( remainder is } \color{blue}{ 15 } \text{ ) } $$

The remainder is still positive ($ 15 > 0 $), so we will continue with division.

Step 6 :
Divide $ 42 $ by $ \color{blue}{ 15 } $ and get the remainder

$$ 42 : 15 = 2 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

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

Step 7 :
Divide $ 15 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 15 : 12 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 8 :
Divide $ 12 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 12 : \color{blue}{ \boxed { 3 }} = 4 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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.

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