Step 1 :
Divide $ 17190 $ by $ 12190 $ and get the remainder

$$ 17190 : 12190 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12190 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12190 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2190 } \text{ ) } $$

The remainder is still positive ($ 2190 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2190 } $ and get the remainder

$$ 5000 : 2190 = 2 \text{ ( remainder is } \color{blue}{ 620 } \text{ ) } $$

The remainder is still positive ($ 620 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2190 $ by $ \color{blue}{ 620 } $ and get the remainder

$$ 2190 : 620 = 3 \text{ ( remainder is } \color{blue}{ 330 } \text{ ) } $$

The remainder is still positive ($ 330 > 0 $), so we will continue with division.

Step 5 :
Divide $ 620 $ by $ \color{blue}{ 330 } $ and get the remainder

$$ 620 : 330 = 1 \text{ ( remainder is } \color{blue}{ 290 } \text{ ) } $$

The remainder is still positive ($ 290 > 0 $), so we will continue with division.

Step 6 :
Divide $ 330 $ by $ \color{blue}{ 290 } $ and get the remainder

$$ 330 : 290 = 1 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

The remainder is still positive ($ 40 > 0 $), so we will continue with division.

Step 7 :
Divide $ 290 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 290 : 40 = 7 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 8 :
Divide $ 40 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 40 : \color{blue}{ \boxed { 10 }} = 4 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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.

GCD Calculator