Step 1 :
Divide $ 17610 $ by $ 12610 $ and get the remainder

$$ 17610 : 12610 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12610 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12610 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2610 } \text{ ) } $$

The remainder is still positive ($ 2610 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2610 } $ and get the remainder

$$ 5000 : 2610 = 1 \text{ ( remainder is } \color{blue}{ 2390 } \text{ ) } $$

The remainder is still positive ($ 2390 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2610 $ by $ \color{blue}{ 2390 } $ and get the remainder

$$ 2610 : 2390 = 1 \text{ ( remainder is } \color{blue}{ 220 } \text{ ) } $$

The remainder is still positive ($ 220 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2390 $ by $ \color{blue}{ 220 } $ and get the remainder

$$ 2390 : 220 = 10 \text{ ( remainder is } \color{blue}{ 190 } \text{ ) } $$

The remainder is still positive ($ 190 > 0 $), so we will continue with division.

Step 6 :
Divide $ 220 $ by $ \color{blue}{ 190 } $ and get the remainder

$$ 220 : 190 = 1 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 7 :
Divide $ 190 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 190 : 30 = 6 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 8 :
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