Step 1 :
Divide $ 17710 $ by $ 12710 $ and get the remainder

$$ 17710 : 12710 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12710 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12710 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2710 } \text{ ) } $$

The remainder is still positive ($ 2710 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2710 } $ and get the remainder

$$ 5000 : 2710 = 1 \text{ ( remainder is } \color{blue}{ 2290 } \text{ ) } $$

The remainder is still positive ($ 2290 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2710 $ by $ \color{blue}{ 2290 } $ and get the remainder

$$ 2710 : 2290 = 1 \text{ ( remainder is } \color{blue}{ 420 } \text{ ) } $$

The remainder is still positive ($ 420 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2290 $ by $ \color{blue}{ 420 } $ and get the remainder

$$ 2290 : 420 = 5 \text{ ( remainder is } \color{blue}{ 190 } \text{ ) } $$

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

Step 6 :
Divide $ 420 $ by $ \color{blue}{ 190 } $ and get the remainder

$$ 420 : 190 = 2 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

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

Step 7 :
Divide $ 190 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 190 : 40 = 4 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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.

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