Step 1 :
Divide $ 688 $ by $ 199 $ and get the remainder

$$ 688 : 199 = 3 \text{ ( remainder is } \color{blue}{ 91 } \text{ ) } $$

The remainder is positive ($ 91 > 0 $), so we will continue with division.

Step 2 :
Divide $ 199 $ by $ \color{blue}{ 91 } $ and get the remainder

$$ 199 : 91 = 2 \text{ ( remainder is } \color{blue}{ 17 } \text{ ) } $$

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

Step 3 :
Divide $ 91 $ by $ \color{blue}{ 17 } $ and get the remainder

$$ 91 : 17 = 5 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 4 :
Divide $ 17 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 17 : 6 = 2 \text{ ( remainder is } \color{blue}{ 5 } \text{ ) } $$

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

Step 5 :
Divide $ 6 $ by $ \color{blue}{ 5 } $ and get the remainder

$$ 6 : 5 = 1 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 6 :
Divide $ 5 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 5 : \color{blue}{ \boxed { 1 }} = 5 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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