Step 1 :
Divide $ 654321 $ by $ 123456 $ and get the remainder

$$ 654321 : 123456 = 5 \text{ ( remainder is } \color{blue}{ 37041 } \text{ ) } $$

The remainder is positive ($ 37041 > 0 $), so we will continue with division.

Step 2 :
Divide $ 123456 $ by $ \color{blue}{ 37041 } $ and get the remainder

$$ 123456 : 37041 = 3 \text{ ( remainder is } \color{blue}{ 12333 } \text{ ) } $$

The remainder is still positive ($ 12333 > 0 $), so we will continue with division.

Step 3 :
Divide $ 37041 $ by $ \color{blue}{ 12333 } $ and get the remainder

$$ 37041 : 12333 = 3 \text{ ( remainder is } \color{blue}{ 42 } \text{ ) } $$

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

Step 4 :
Divide $ 12333 $ by $ \color{blue}{ 42 } $ and get the remainder

$$ 12333 : 42 = 293 \text{ ( remainder is } \color{blue}{ 27 } \text{ ) } $$

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

Step 5 :
Divide $ 42 $ by $ \color{blue}{ 27 } $ and get the remainder

$$ 42 : 27 = 1 \text{ ( remainder is } \color{blue}{ 15 } \text{ ) } $$

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

Step 6 :
Divide $ 27 $ by $ \color{blue}{ 15 } $ and get the remainder

$$ 27 : 15 = 1 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

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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