Step 1 :
Divide $ 17870 $ by $ 12870 $ and get the remainder

$$ 17870 : 12870 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12870 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12870 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2870 } \text{ ) } $$

The remainder is still positive ($ 2870 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2870 } $ and get the remainder

$$ 5000 : 2870 = 1 \text{ ( remainder is } \color{blue}{ 2130 } \text{ ) } $$

The remainder is still positive ($ 2130 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2870 $ by $ \color{blue}{ 2130 } $ and get the remainder

$$ 2870 : 2130 = 1 \text{ ( remainder is } \color{blue}{ 740 } \text{ ) } $$

The remainder is still positive ($ 740 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2130 $ by $ \color{blue}{ 740 } $ and get the remainder

$$ 2130 : 740 = 2 \text{ ( remainder is } \color{blue}{ 650 } \text{ ) } $$

The remainder is still positive ($ 650 > 0 $), so we will continue with division.

Step 6 :
Divide $ 740 $ by $ \color{blue}{ 650 } $ and get the remainder

$$ 740 : 650 = 1 \text{ ( remainder is } \color{blue}{ 90 } \text{ ) } $$

The remainder is still positive ($ 90 > 0 $), so we will continue with division.

Step 7 :
Divide $ 650 $ by $ \color{blue}{ 90 } $ and get the remainder

$$ 650 : 90 = 7 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

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

Step 8 :
Divide $ 90 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 90 : 20 = 4 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 9 :
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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