Step 1 :
Divide $ 1071 $ by $ 588 $ and get the remainder

$$ 1071 : 588 = 1 \text{ ( remainder is } \color{blue}{ 483 } \text{ ) } $$

The remainder is positive ($ 483 > 0 $), so we will continue with division.

Step 2 :
Divide $ 588 $ by $ \color{blue}{ 483 } $ and get the remainder

$$ 588 : 483 = 1 \text{ ( remainder is } \color{blue}{ 105 } \text{ ) } $$

The remainder is still positive ($ 105 > 0 $), so we will continue with division.

Step 3 :
Divide $ 483 $ by $ \color{blue}{ 105 } $ and get the remainder

$$ 483 : 105 = 4 \text{ ( remainder is } \color{blue}{ 63 } \text{ ) } $$

The remainder is still positive ($ 63 > 0 $), so we will continue with division.

Step 4 :
Divide $ 105 $ by $ \color{blue}{ 63 } $ and get the remainder

$$ 105 : 63 = 1 \text{ ( remainder is } \color{blue}{ 42 } \text{ ) } $$

The remainder is still positive ($ 42 > 0 $), so we will continue with division.

Step 5 :
Divide $ 63 $ by $ \color{blue}{ 42 } $ and get the remainder

$$ 63 : 42 = 1 \text{ ( remainder is } \color{blue}{ 21 } \text{ ) } $$

The remainder is still positive ($ 21 > 0 $), so we will continue with division.

Step 6 :
Divide $ 42 $ by $ \color{blue}{ 21 } $ and get the remainder

$$ 42 : \color{blue}{ \boxed { 21 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 21 }} $.

This solution can be visualized using a Venn diagram.

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

GCD Calculator