Step 1 :
Divide $ 15270 $ by $ 10270 $ and get the remainder

$$ 15270 : 10270 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 10270 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 10270 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 270 } \text{ ) } $$

The remainder is still positive ($ 270 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 270 } $ and get the remainder

$$ 5000 : 270 = 18 \text{ ( remainder is } \color{blue}{ 140 } \text{ ) } $$

The remainder is still positive ($ 140 > 0 $), so we will continue with division.

Step 4 :
Divide $ 270 $ by $ \color{blue}{ 140 } $ and get the remainder

$$ 270 : 140 = 1 \text{ ( remainder is } \color{blue}{ 130 } \text{ ) } $$

The remainder is still positive ($ 130 > 0 $), so we will continue with division.

Step 5 :
Divide $ 140 $ by $ \color{blue}{ 130 } $ and get the remainder

$$ 140 : 130 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 6 :
Divide $ 130 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 130 : \color{blue}{ \boxed { 10 }} = 13 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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