Step 1 :
Divide $ 2025 $ by $ 1588 $ and get the remainder

$$ 2025 : 1588 = 1 \text{ ( remainder is } \color{blue}{ 437 } \text{ ) } $$

The remainder is positive ($ 437 > 0 $), so we will continue with division.

Step 2 :
Divide $ 1588 $ by $ \color{blue}{ 437 } $ and get the remainder

$$ 1588 : 437 = 3 \text{ ( remainder is } \color{blue}{ 277 } \text{ ) } $$

The remainder is still positive ($ 277 > 0 $), so we will continue with division.

Step 3 :
Divide $ 437 $ by $ \color{blue}{ 277 } $ and get the remainder

$$ 437 : 277 = 1 \text{ ( remainder is } \color{blue}{ 160 } \text{ ) } $$

The remainder is still positive ($ 160 > 0 $), so we will continue with division.

Step 4 :
Divide $ 277 $ by $ \color{blue}{ 160 } $ and get the remainder

$$ 277 : 160 = 1 \text{ ( remainder is } \color{blue}{ 117 } \text{ ) } $$

The remainder is still positive ($ 117 > 0 $), so we will continue with division.

Step 5 :
Divide $ 160 $ by $ \color{blue}{ 117 } $ and get the remainder

$$ 160 : 117 = 1 \text{ ( remainder is } \color{blue}{ 43 } \text{ ) } $$

The remainder is still positive ($ 43 > 0 $), so we will continue with division.

Step 6 :
Divide $ 117 $ by $ \color{blue}{ 43 } $ and get the remainder

$$ 117 : 43 = 2 \text{ ( remainder is } \color{blue}{ 31 } \text{ ) } $$

The remainder is still positive ($ 31 > 0 $), so we will continue with division.

Step 7 :
Divide $ 43 $ by $ \color{blue}{ 31 } $ and get the remainder

$$ 43 : 31 = 1 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

The remainder is still positive ($ 12 > 0 $), so we will continue with division.

Step 8 :
Divide $ 31 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 31 : 12 = 2 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

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

Step 9 :
Divide $ 12 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 12 : 7 = 1 \text{ ( remainder is } \color{blue}{ 5 } \text{ ) } $$

The remainder is still positive ($ 5 > 0 $), so we will continue with division.

Step 10 :
Divide $ 7 $ by $ \color{blue}{ 5 } $ and get the remainder

$$ 7 : 5 = 1 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

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

Step 11 :
Divide $ 5 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 5 : 2 = 2 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 12 :
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.

GCD Calculator