Step 1 :
Divide $ 12345 $ by $ 8991 $ and get the remainder

$$ 12345 : 8991 = 1 \text{ ( remainder is } \color{blue}{ 3354 } \text{ ) } $$

The remainder is positive ($ 3354 > 0 $), so we will continue with division.

Step 2 :
Divide $ 8991 $ by $ \color{blue}{ 3354 } $ and get the remainder

$$ 8991 : 3354 = 2 \text{ ( remainder is } \color{blue}{ 2283 } \text{ ) } $$

The remainder is still positive ($ 2283 > 0 $), so we will continue with division.

Step 3 :
Divide $ 3354 $ by $ \color{blue}{ 2283 } $ and get the remainder

$$ 3354 : 2283 = 1 \text{ ( remainder is } \color{blue}{ 1071 } \text{ ) } $$

The remainder is still positive ($ 1071 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2283 $ by $ \color{blue}{ 1071 } $ and get the remainder

$$ 2283 : 1071 = 2 \text{ ( remainder is } \color{blue}{ 141 } \text{ ) } $$

The remainder is still positive ($ 141 > 0 $), so we will continue with division.

Step 5 :
Divide $ 1071 $ by $ \color{blue}{ 141 } $ and get the remainder

$$ 1071 : 141 = 7 \text{ ( remainder is } \color{blue}{ 84 } \text{ ) } $$

The remainder is still positive ($ 84 > 0 $), so we will continue with division.

Step 6 :
Divide $ 141 $ by $ \color{blue}{ 84 } $ and get the remainder

$$ 141 : 84 = 1 \text{ ( remainder is } \color{blue}{ 57 } \text{ ) } $$

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

Step 7 :
Divide $ 84 $ by $ \color{blue}{ 57 } $ and get the remainder

$$ 84 : 57 = 1 \text{ ( remainder is } \color{blue}{ 27 } \text{ ) } $$

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

Step 8 :
Divide $ 57 $ by $ \color{blue}{ 27 } $ and get the remainder

$$ 57 : 27 = 2 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 9 :
Divide $ 27 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 27 : \color{blue}{ \boxed { 3 }} = 9 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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