Step 1 :
Divide $ 12345 $ by $ 5040 $ and get the remainder

$$ 12345 : 5040 = 2 \text{ ( remainder is } \color{blue}{ 2265 } \text{ ) } $$

The remainder is positive ($ 2265 > 0 $), so we will continue with division.

Step 2 :
Divide $ 5040 $ by $ \color{blue}{ 2265 } $ and get the remainder

$$ 5040 : 2265 = 2 \text{ ( remainder is } \color{blue}{ 510 } \text{ ) } $$

The remainder is still positive ($ 510 > 0 $), so we will continue with division.

Step 3 :
Divide $ 2265 $ by $ \color{blue}{ 510 } $ and get the remainder

$$ 2265 : 510 = 4 \text{ ( remainder is } \color{blue}{ 225 } \text{ ) } $$

The remainder is still positive ($ 225 > 0 $), so we will continue with division.

Step 4 :
Divide $ 510 $ by $ \color{blue}{ 225 } $ and get the remainder

$$ 510 : 225 = 2 \text{ ( remainder is } \color{blue}{ 60 } \text{ ) } $$

The remainder is still positive ($ 60 > 0 $), so we will continue with division.

Step 5 :
Divide $ 225 $ by $ \color{blue}{ 60 } $ and get the remainder

$$ 225 : 60 = 3 \text{ ( remainder is } \color{blue}{ 45 } \text{ ) } $$

The remainder is still positive ($ 45 > 0 $), so we will continue with division.

Step 6 :
Divide $ 60 $ by $ \color{blue}{ 45 } $ and get the remainder

$$ 60 : 45 = 1 \text{ ( remainder is } \color{blue}{ 15 } \text{ ) } $$

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

Step 7 :
Divide $ 45 $ by $ \color{blue}{ 15 } $ and get the remainder

$$ 45 : \color{blue}{ \boxed { 15 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 15 }} $.

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

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