Step 1 :
Divide $ 18170 $ by $ 13170 $ and get the remainder

$$ 18170 : 13170 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 13170 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 13170 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 3170 } \text{ ) } $$

The remainder is still positive ($ 3170 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 3170 } $ and get the remainder

$$ 5000 : 3170 = 1 \text{ ( remainder is } \color{blue}{ 1830 } \text{ ) } $$

The remainder is still positive ($ 1830 > 0 $), so we will continue with division.

Step 4 :
Divide $ 3170 $ by $ \color{blue}{ 1830 } $ and get the remainder

$$ 3170 : 1830 = 1 \text{ ( remainder is } \color{blue}{ 1340 } \text{ ) } $$

The remainder is still positive ($ 1340 > 0 $), so we will continue with division.

Step 5 :
Divide $ 1830 $ by $ \color{blue}{ 1340 } $ and get the remainder

$$ 1830 : 1340 = 1 \text{ ( remainder is } \color{blue}{ 490 } \text{ ) } $$

The remainder is still positive ($ 490 > 0 $), so we will continue with division.

Step 6 :
Divide $ 1340 $ by $ \color{blue}{ 490 } $ and get the remainder

$$ 1340 : 490 = 2 \text{ ( remainder is } \color{blue}{ 360 } \text{ ) } $$

The remainder is still positive ($ 360 > 0 $), so we will continue with division.

Step 7 :
Divide $ 490 $ by $ \color{blue}{ 360 } $ and get the remainder

$$ 490 : 360 = 1 \text{ ( remainder is } \color{blue}{ 130 } \text{ ) } $$

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

Step 8 :
Divide $ 360 $ by $ \color{blue}{ 130 } $ and get the remainder

$$ 360 : 130 = 2 \text{ ( remainder is } \color{blue}{ 100 } \text{ ) } $$

The remainder is still positive ($ 100 > 0 $), so we will continue with division.

Step 9 :
Divide $ 130 $ by $ \color{blue}{ 100 } $ and get the remainder

$$ 130 : 100 = 1 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 10 :
Divide $ 100 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 100 : 30 = 3 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 11 :
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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