Step 1 :
Divide $ 17590 $ by $ 12590 $ and get the remainder

$$ 17590 : 12590 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12590 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12590 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2590 } \text{ ) } $$

The remainder is still positive ($ 2590 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2590 } $ and get the remainder

$$ 5000 : 2590 = 1 \text{ ( remainder is } \color{blue}{ 2410 } \text{ ) } $$

The remainder is still positive ($ 2410 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2590 $ by $ \color{blue}{ 2410 } $ and get the remainder

$$ 2590 : 2410 = 1 \text{ ( remainder is } \color{blue}{ 180 } \text{ ) } $$

The remainder is still positive ($ 180 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2410 $ by $ \color{blue}{ 180 } $ and get the remainder

$$ 2410 : 180 = 13 \text{ ( remainder is } \color{blue}{ 70 } \text{ ) } $$

The remainder is still positive ($ 70 > 0 $), so we will continue with division.

Step 6 :
Divide $ 180 $ by $ \color{blue}{ 70 } $ and get the remainder

$$ 180 : 70 = 2 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

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

Step 7 :
Divide $ 70 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 70 : 40 = 1 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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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