Step 1 :
Divide $ 1999 $ by $ 299 $ and get the remainder

$$ 1999 : 299 = 6 \text{ ( remainder is } \color{blue}{ 205 } \text{ ) } $$

The remainder is positive ($ 205 > 0 $), so we will continue with division.

Step 2 :
Divide $ 299 $ by $ \color{blue}{ 205 } $ and get the remainder

$$ 299 : 205 = 1 \text{ ( remainder is } \color{blue}{ 94 } \text{ ) } $$

The remainder is still positive ($ 94 > 0 $), so we will continue with division.

Step 3 :
Divide $ 205 $ by $ \color{blue}{ 94 } $ and get the remainder

$$ 205 : 94 = 2 \text{ ( remainder is } \color{blue}{ 17 } \text{ ) } $$

The remainder is still positive ($ 17 > 0 $), so we will continue with division.

Step 4 :
Divide $ 94 $ by $ \color{blue}{ 17 } $ and get the remainder

$$ 94 : 17 = 5 \text{ ( remainder is } \color{blue}{ 9 } \text{ ) } $$

The remainder is still positive ($ 9 > 0 $), so we will continue with division.

Step 5 :
Divide $ 17 $ by $ \color{blue}{ 9 } $ and get the remainder

$$ 17 : 9 = 1 \text{ ( remainder is } \color{blue}{ 8 } \text{ ) } $$

The remainder is still positive ($ 8 > 0 $), so we will continue with division.

Step 6 :
Divide $ 9 $ by $ \color{blue}{ 8 } $ and get the remainder

$$ 9 : 8 = 1 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 7 :
Divide $ 8 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 8 : \color{blue}{ \boxed { 1 }} = 8 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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