Step 1 :
Divide $ 15170 $ by $ 10170 $ and get the remainder

$$ 15170 : 10170 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 10170 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 10170 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 170 } \text{ ) } $$

The remainder is still positive ($ 170 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 170 } $ and get the remainder

$$ 5000 : 170 = 29 \text{ ( remainder is } \color{blue}{ 70 } \text{ ) } $$

The remainder is still positive ($ 70 > 0 $), so we will continue with division.

Step 4 :
Divide $ 170 $ by $ \color{blue}{ 70 } $ and get the remainder

$$ 170 : 70 = 2 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 5 :
Divide $ 70 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 70 : 30 = 2 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 6 :
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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