Step 1 :
Divide $ 17270 $ by $ 12270 $ and get the remainder

$$ 17270 : 12270 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12270 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12270 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2270 } \text{ ) } $$

The remainder is still positive ($ 2270 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2270 } $ and get the remainder

$$ 5000 : 2270 = 2 \text{ ( remainder is } \color{blue}{ 460 } \text{ ) } $$

The remainder is still positive ($ 460 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2270 $ by $ \color{blue}{ 460 } $ and get the remainder

$$ 2270 : 460 = 4 \text{ ( remainder is } \color{blue}{ 430 } \text{ ) } $$

The remainder is still positive ($ 430 > 0 $), so we will continue with division.

Step 5 :
Divide $ 460 $ by $ \color{blue}{ 430 } $ and get the remainder

$$ 460 : 430 = 1 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 6 :
Divide $ 430 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 430 : 30 = 14 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 7 :
Divide $ 30 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 30 : \color{blue}{ \boxed { 10 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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