Step 1 :
Divide $ 558 $ by $ 220 $ and get the remainder

$$ 558 : 220 = 2 \text{ ( remainder is } \color{blue}{ 118 } \text{ ) } $$

The remainder is positive ($ 118 > 0 $), so we will continue with division.

Step 2 :
Divide $ 220 $ by $ \color{blue}{ 118 } $ and get the remainder

$$ 220 : 118 = 1 \text{ ( remainder is } \color{blue}{ 102 } \text{ ) } $$

The remainder is still positive ($ 102 > 0 $), so we will continue with division.

Step 3 :
Divide $ 118 $ by $ \color{blue}{ 102 } $ and get the remainder

$$ 118 : 102 = 1 \text{ ( remainder is } \color{blue}{ 16 } \text{ ) } $$

The remainder is still positive ($ 16 > 0 $), so we will continue with division.

Step 4 :
Divide $ 102 $ by $ \color{blue}{ 16 } $ and get the remainder

$$ 102 : 16 = 6 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 5 :
Divide $ 16 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 16 : 6 = 2 \text{ ( remainder is } \color{blue}{ 4 } \text{ ) } $$

The remainder is still positive ($ 4 > 0 $), so we will continue with division.

Step 6 :
Divide $ 6 $ by $ \color{blue}{ 4 } $ and get the remainder

$$ 6 : 4 = 1 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

The remainder is still positive ($ 2 > 0 $), so we will continue with division.

Step 7 :
Divide $ 4 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 4 : \color{blue}{ \boxed { 2 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 2 }} $.

This solution can be visualized using a Venn diagram.

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

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