Step 1 :
Divide $ 17970 $ by $ 12970 $ and get the remainder

$$ 17970 : 12970 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12970 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12970 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2970 } \text{ ) } $$

The remainder is still positive ($ 2970 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2970 } $ and get the remainder

$$ 5000 : 2970 = 1 \text{ ( remainder is } \color{blue}{ 2030 } \text{ ) } $$

The remainder is still positive ($ 2030 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2970 $ by $ \color{blue}{ 2030 } $ and get the remainder

$$ 2970 : 2030 = 1 \text{ ( remainder is } \color{blue}{ 940 } \text{ ) } $$

The remainder is still positive ($ 940 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2030 $ by $ \color{blue}{ 940 } $ and get the remainder

$$ 2030 : 940 = 2 \text{ ( remainder is } \color{blue}{ 150 } \text{ ) } $$

The remainder is still positive ($ 150 > 0 $), so we will continue with division.

Step 6 :
Divide $ 940 $ by $ \color{blue}{ 150 } $ and get the remainder

$$ 940 : 150 = 6 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

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

Step 7 :
Divide $ 150 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 150 : 40 = 3 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 8 :
Divide $ 40 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 40 : 30 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 9 :
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.

GCD Calculator