Step 1 :
Divide $ 83 $ by $ 51 $ and get the remainder

$$ 83 : 51 = 1 \text{ ( remainder is } \color{blue}{ 32 } \text{ ) } $$

The remainder is positive ($ 32 > 0 $), so we will continue with division.

Step 2 :
Divide $ 51 $ by $ \color{blue}{ 32 } $ and get the remainder

$$ 51 : 32 = 1 \text{ ( remainder is } \color{blue}{ 19 } \text{ ) } $$

The remainder is still positive ($ 19 > 0 $), so we will continue with division.

Step 3 :
Divide $ 32 $ by $ \color{blue}{ 19 } $ and get the remainder

$$ 32 : 19 = 1 \text{ ( remainder is } \color{blue}{ 13 } \text{ ) } $$

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

Step 4 :
Divide $ 19 $ by $ \color{blue}{ 13 } $ and get the remainder

$$ 19 : 13 = 1 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

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

Step 5 :
Divide $ 13 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 13 : 6 = 2 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 6 :
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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