Step 1 :
Divide $ 18270 $ by $ 13270 $ and get the remainder

$$ 18270 : 13270 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 13270 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 13270 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 3270 } \text{ ) } $$

The remainder is still positive ($ 3270 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 3270 } $ and get the remainder

$$ 5000 : 3270 = 1 \text{ ( remainder is } \color{blue}{ 1730 } \text{ ) } $$

The remainder is still positive ($ 1730 > 0 $), so we will continue with division.

Step 4 :
Divide $ 3270 $ by $ \color{blue}{ 1730 } $ and get the remainder

$$ 3270 : 1730 = 1 \text{ ( remainder is } \color{blue}{ 1540 } \text{ ) } $$

The remainder is still positive ($ 1540 > 0 $), so we will continue with division.

Step 5 :
Divide $ 1730 $ by $ \color{blue}{ 1540 } $ and get the remainder

$$ 1730 : 1540 = 1 \text{ ( remainder is } \color{blue}{ 190 } \text{ ) } $$

The remainder is still positive ($ 190 > 0 $), so we will continue with division.

Step 6 :
Divide $ 1540 $ by $ \color{blue}{ 190 } $ and get the remainder

$$ 1540 : 190 = 8 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

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

Step 7 :
Divide $ 190 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 190 : 20 = 9 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 8 :
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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