Step 1 :
Divide $ 583 $ by $ 217 $ and get the remainder

$$ 583 : 217 = 2 \text{ ( remainder is } \color{blue}{ 149 } \text{ ) } $$

The remainder is positive ($ 149 > 0 $), so we will continue with division.

Step 2 :
Divide $ 217 $ by $ \color{blue}{ 149 } $ and get the remainder

$$ 217 : 149 = 1 \text{ ( remainder is } \color{blue}{ 68 } \text{ ) } $$

The remainder is still positive ($ 68 > 0 $), so we will continue with division.

Step 3 :
Divide $ 149 $ by $ \color{blue}{ 68 } $ and get the remainder

$$ 149 : 68 = 2 \text{ ( remainder is } \color{blue}{ 13 } \text{ ) } $$

The remainder is still positive ($ 13 > 0 $), so we will continue with division.

Step 4 :
Divide $ 68 $ by $ \color{blue}{ 13 } $ and get the remainder

$$ 68 : 13 = 5 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

The remainder is still positive ($ 3 > 0 $), so we will continue with division.

Step 5 :
Divide $ 13 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 13 : 3 = 4 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 6 :
Divide $ 3 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 3 : \color{blue}{ \boxed { 1 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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