Step 1 :
Divide $ 17910 $ by $ 12910 $ and get the remainder

$$ 17910 : 12910 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12910 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12910 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2910 } \text{ ) } $$

The remainder is still positive ($ 2910 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2910 } $ and get the remainder

$$ 5000 : 2910 = 1 \text{ ( remainder is } \color{blue}{ 2090 } \text{ ) } $$

The remainder is still positive ($ 2090 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2910 $ by $ \color{blue}{ 2090 } $ and get the remainder

$$ 2910 : 2090 = 1 \text{ ( remainder is } \color{blue}{ 820 } \text{ ) } $$

The remainder is still positive ($ 820 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2090 $ by $ \color{blue}{ 820 } $ and get the remainder

$$ 2090 : 820 = 2 \text{ ( remainder is } \color{blue}{ 450 } \text{ ) } $$

The remainder is still positive ($ 450 > 0 $), so we will continue with division.

Step 6 :
Divide $ 820 $ by $ \color{blue}{ 450 } $ and get the remainder

$$ 820 : 450 = 1 \text{ ( remainder is } \color{blue}{ 370 } \text{ ) } $$

The remainder is still positive ($ 370 > 0 $), so we will continue with division.

Step 7 :
Divide $ 450 $ by $ \color{blue}{ 370 } $ and get the remainder

$$ 450 : 370 = 1 \text{ ( remainder is } \color{blue}{ 80 } \text{ ) } $$

The remainder is still positive ($ 80 > 0 $), so we will continue with division.

Step 8 :
Divide $ 370 $ by $ \color{blue}{ 80 } $ and get the remainder

$$ 370 : 80 = 4 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

The remainder is still positive ($ 50 > 0 $), so we will continue with division.

Step 9 :
Divide $ 80 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 80 : 50 = 1 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 10 :
Divide $ 50 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 50 : 30 = 1 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

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

Step 11 :
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 12 :
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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