Step 1 :
Divide $ 856641 $ by $ 587877 $ and get the remainder

$$ 856641 : 587877 = 1 \text{ ( remainder is } \color{blue}{ 268764 } \text{ ) } $$

The remainder is positive ($ 268764 > 0 $), so we will continue with division.

Step 2 :
Divide $ 587877 $ by $ \color{blue}{ 268764 } $ and get the remainder

$$ 587877 : 268764 = 2 \text{ ( remainder is } \color{blue}{ 50349 } \text{ ) } $$

The remainder is still positive ($ 50349 > 0 $), so we will continue with division.

Step 3 :
Divide $ 268764 $ by $ \color{blue}{ 50349 } $ and get the remainder

$$ 268764 : 50349 = 5 \text{ ( remainder is } \color{blue}{ 17019 } \text{ ) } $$

The remainder is still positive ($ 17019 > 0 $), so we will continue with division.

Step 4 :
Divide $ 50349 $ by $ \color{blue}{ 17019 } $ and get the remainder

$$ 50349 : 17019 = 2 \text{ ( remainder is } \color{blue}{ 16311 } \text{ ) } $$

The remainder is still positive ($ 16311 > 0 $), so we will continue with division.

Step 5 :
Divide $ 17019 $ by $ \color{blue}{ 16311 } $ and get the remainder

$$ 17019 : 16311 = 1 \text{ ( remainder is } \color{blue}{ 708 } \text{ ) } $$

The remainder is still positive ($ 708 > 0 $), so we will continue with division.

Step 6 :
Divide $ 16311 $ by $ \color{blue}{ 708 } $ and get the remainder

$$ 16311 : 708 = 23 \text{ ( remainder is } \color{blue}{ 27 } \text{ ) } $$

The remainder is still positive ($ 27 > 0 $), so we will continue with division.

Step 7 :
Divide $ 708 $ by $ \color{blue}{ 27 } $ and get the remainder

$$ 708 : 27 = 26 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 8 :
Divide $ 27 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 27 : 6 = 4 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 9 :
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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