Step 1 :
Divide $ 17510 $ by $ 12510 $ and get the remainder

$$ 17510 : 12510 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12510 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12510 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2510 } \text{ ) } $$

The remainder is still positive ($ 2510 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2510 } $ and get the remainder

$$ 5000 : 2510 = 1 \text{ ( remainder is } \color{blue}{ 2490 } \text{ ) } $$

The remainder is still positive ($ 2490 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2510 $ by $ \color{blue}{ 2490 } $ and get the remainder

$$ 2510 : 2490 = 1 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

The remainder is still positive ($ 20 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2490 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 2490 : 20 = 124 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 6 :
Divide $ 20 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 20 : \color{blue}{ \boxed { 10 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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