Step 1 :
Divide $ 15590 $ by $ 10590 $ and get the remainder

$$ 15590 : 10590 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 10590 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 10590 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 590 } \text{ ) } $$

The remainder is still positive ($ 590 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 590 } $ and get the remainder

$$ 5000 : 590 = 8 \text{ ( remainder is } \color{blue}{ 280 } \text{ ) } $$

The remainder is still positive ($ 280 > 0 $), so we will continue with division.

Step 4 :
Divide $ 590 $ by $ \color{blue}{ 280 } $ and get the remainder

$$ 590 : 280 = 2 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

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

Step 5 :
Divide $ 280 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 280 : 30 = 9 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 6 :
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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