Step 1 :
Divide $ 786 $ by $ 123 $ and get the remainder

$$ 786 : 123 = 6 \text{ ( remainder is } \color{blue}{ 48 } \text{ ) } $$

The remainder is positive ($ 48 > 0 $), so we will continue with division.

Step 2 :
Divide $ 123 $ by $ \color{blue}{ 48 } $ and get the remainder

$$ 123 : 48 = 2 \text{ ( remainder is } \color{blue}{ 27 } \text{ ) } $$

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

Step 3 :
Divide $ 48 $ by $ \color{blue}{ 27 } $ and get the remainder

$$ 48 : 27 = 1 \text{ ( remainder is } \color{blue}{ 21 } \text{ ) } $$

The remainder is still positive ($ 21 > 0 $), so we will continue with division.

Step 4 :
Divide $ 27 $ by $ \color{blue}{ 21 } $ and get the remainder

$$ 27 : 21 = 1 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 5 :
Divide $ 21 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 21 : 6 = 3 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 6 :
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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