Step 1 :
Divide $ 19999 $ by $ 299 $ and get the remainder

$$ 19999 : 299 = 66 \text{ ( remainder is } \color{blue}{ 265 } \text{ ) } $$

The remainder is positive ($ 265 > 0 $), so we will continue with division.

Step 2 :
Divide $ 299 $ by $ \color{blue}{ 265 } $ and get the remainder

$$ 299 : 265 = 1 \text{ ( remainder is } \color{blue}{ 34 } \text{ ) } $$

The remainder is still positive ($ 34 > 0 $), so we will continue with division.

Step 3 :
Divide $ 265 $ by $ \color{blue}{ 34 } $ and get the remainder

$$ 265 : 34 = 7 \text{ ( remainder is } \color{blue}{ 27 } \text{ ) } $$

The remainder is still positive ($ 27 > 0 $), so we will continue with division.

Step 4 :
Divide $ 34 $ by $ \color{blue}{ 27 } $ and get the remainder

$$ 34 : 27 = 1 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

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

Step 5 :
Divide $ 27 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 27 : 7 = 3 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

The remainder is still positive ($ 6 > 0 $), so we will continue with division.

Step 6 :
Divide $ 7 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 7 : 6 = 1 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 7 :
Divide $ 6 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 6 : \color{blue}{ \boxed { 1 }} = 6 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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