Step 1 :
Divide $ 500000001 $ by $ 5000001 $ and get the remainder

$$ 500000001 : 5000001 = 99 \text{ ( remainder is } \color{blue}{ 4999902 } \text{ ) } $$

The remainder is positive ($ 4999902 > 0 $), so we will continue with division.

Step 2 :
Divide $ 5000001 $ by $ \color{blue}{ 4999902 } $ and get the remainder

$$ 5000001 : 4999902 = 1 \text{ ( remainder is } \color{blue}{ 99 } \text{ ) } $$

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

Step 3 :
Divide $ 4999902 $ by $ \color{blue}{ 99 } $ and get the remainder

$$ 4999902 : 99 = 50504 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 4 :
Divide $ 99 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 99 : 6 = 16 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 5 :
Divide $ 6 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 6 : \color{blue}{ \boxed { 3 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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