Step 1 :
Divide $ 421319712 $ by $ 271566 $ and get the remainder

$$ 421319712 : 271566 = 1551 \text{ ( remainder is } \color{blue}{ 120846 } \text{ ) } $$

The remainder is positive ($ 120846 > 0 $), so we will continue with division.

Step 2 :
Divide $ 271566 $ by $ \color{blue}{ 120846 } $ and get the remainder

$$ 271566 : 120846 = 2 \text{ ( remainder is } \color{blue}{ 29874 } \text{ ) } $$

The remainder is still positive ($ 29874 > 0 $), so we will continue with division.

Step 3 :
Divide $ 120846 $ by $ \color{blue}{ 29874 } $ and get the remainder

$$ 120846 : 29874 = 4 \text{ ( remainder is } \color{blue}{ 1350 } \text{ ) } $$

The remainder is still positive ($ 1350 > 0 $), so we will continue with division.

Step 4 :
Divide $ 29874 $ by $ \color{blue}{ 1350 } $ and get the remainder

$$ 29874 : 1350 = 22 \text{ ( remainder is } \color{blue}{ 174 } \text{ ) } $$

The remainder is still positive ($ 174 > 0 $), so we will continue with division.

Step 5 :
Divide $ 1350 $ by $ \color{blue}{ 174 } $ and get the remainder

$$ 1350 : 174 = 7 \text{ ( remainder is } \color{blue}{ 132 } \text{ ) } $$

The remainder is still positive ($ 132 > 0 $), so we will continue with division.

Step 6 :
Divide $ 174 $ by $ \color{blue}{ 132 } $ and get the remainder

$$ 174 : 132 = 1 \text{ ( remainder is } \color{blue}{ 42 } \text{ ) } $$

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

Step 7 :
Divide $ 132 $ by $ \color{blue}{ 42 } $ and get the remainder

$$ 132 : 42 = 3 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 8 :
Divide $ 42 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 42 : \color{blue}{ \boxed { 6 }} = 7 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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