Step 1 :
Divide $ 17790 $ by $ 12790 $ and get the remainder

$$ 17790 : 12790 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

The remainder is positive ($ 5000 > 0 $), so we will continue with division.

Step 2 :
Divide $ 12790 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12790 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2790 } \text{ ) } $$

The remainder is still positive ($ 2790 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2790 } $ and get the remainder

$$ 5000 : 2790 = 1 \text{ ( remainder is } \color{blue}{ 2210 } \text{ ) } $$

The remainder is still positive ($ 2210 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2790 $ by $ \color{blue}{ 2210 } $ and get the remainder

$$ 2790 : 2210 = 1 \text{ ( remainder is } \color{blue}{ 580 } \text{ ) } $$

The remainder is still positive ($ 580 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2210 $ by $ \color{blue}{ 580 } $ and get the remainder

$$ 2210 : 580 = 3 \text{ ( remainder is } \color{blue}{ 470 } \text{ ) } $$

The remainder is still positive ($ 470 > 0 $), so we will continue with division.

Step 6 :
Divide $ 580 $ by $ \color{blue}{ 470 } $ and get the remainder

$$ 580 : 470 = 1 \text{ ( remainder is } \color{blue}{ 110 } \text{ ) } $$

The remainder is still positive ($ 110 > 0 $), so we will continue with division.

Step 7 :
Divide $ 470 $ by $ \color{blue}{ 110 } $ and get the remainder

$$ 470 : 110 = 4 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

The remainder is still positive ($ 30 > 0 $), so we will continue with division.

Step 8 :
Divide $ 110 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 110 : 30 = 3 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

The remainder is still positive ($ 20 > 0 $), so we will continue with division.

Step 9 :
Divide $ 30 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 30 : 20 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

The remainder is still positive ($ 10 > 0 $), so we will continue with division.

Step 10 :
Divide $ 20 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 20 : \color{blue}{ \boxed { 10 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 10 }} $.

This solution can be visualized using a Venn diagram.

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

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