Step 1 :
Divide $ 2592 $ by $ 2569 $ and get the remainder

$$ 2592 : 2569 = 1 \text{ ( remainder is } \color{blue}{ 23 } \text{ ) } $$

The remainder is positive ($ 23 > 0 $), so we will continue with division.

Step 2 :
Divide $ 2569 $ by $ \color{blue}{ 23 } $ and get the remainder

$$ 2569 : 23 = 111 \text{ ( remainder is } \color{blue}{ 16 } \text{ ) } $$

The remainder is still positive ($ 16 > 0 $), so we will continue with division.

Step 3 :
Divide $ 23 $ by $ \color{blue}{ 16 } $ and get the remainder

$$ 23 : 16 = 1 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

The remainder is still positive ($ 7 > 0 $), so we will continue with division.

Step 4 :
Divide $ 16 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 16 : 7 = 2 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

The remainder is still positive ($ 2 > 0 $), so we will continue with division.

Step 5 :
Divide $ 7 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 7 : 2 = 3 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

The remainder is still positive ($ 1 > 0 $), so we will continue with division.

Step 6 :
Divide $ 2 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 2 : \color{blue}{ \boxed { 1 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 1 }} $.

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

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