Step 1 :
Divide $ 6123 $ by $ 2913 $ and get the remainder

$$ 6123 : 2913 = 2 \text{ ( remainder is } \color{blue}{ 297 } \text{ ) } $$

The remainder is positive ($ 297 > 0 $), so we will continue with division.

Step 2 :
Divide $ 2913 $ by $ \color{blue}{ 297 } $ and get the remainder

$$ 2913 : 297 = 9 \text{ ( remainder is } \color{blue}{ 240 } \text{ ) } $$

The remainder is still positive ($ 240 > 0 $), so we will continue with division.

Step 3 :
Divide $ 297 $ by $ \color{blue}{ 240 } $ and get the remainder

$$ 297 : 240 = 1 \text{ ( remainder is } \color{blue}{ 57 } \text{ ) } $$

The remainder is still positive ($ 57 > 0 $), so we will continue with division.

Step 4 :
Divide $ 240 $ by $ \color{blue}{ 57 } $ and get the remainder

$$ 240 : 57 = 4 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

The remainder is still positive ($ 12 > 0 $), so we will continue with division.

Step 5 :
Divide $ 57 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 57 : 12 = 4 \text{ ( remainder is } \color{blue}{ 9 } \text{ ) } $$

The remainder is still positive ($ 9 > 0 $), so we will continue with division.

Step 6 :
Divide $ 12 $ by $ \color{blue}{ 9 } $ and get the remainder

$$ 12 : 9 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 7 :
Divide $ 9 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 9 : \color{blue}{ \boxed { 3 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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