Step 1 :
Divide $ 623 $ by $ 51 $ and get the remainder

$$ 623 : 51 = 12 \text{ ( remainder is } \color{blue}{ 11 } \text{ ) } $$

The remainder is positive ($ 11 > 0 $), so we will continue with division.

Step 2 :
Divide $ 51 $ by $ \color{blue}{ 11 } $ and get the remainder

$$ 51 : 11 = 4 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

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

Step 3 :
Divide $ 11 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 11 : 7 = 1 \text{ ( remainder is } \color{blue}{ 4 } \text{ ) } $$

The remainder is still positive ($ 4 > 0 $), so we will continue with division.

Step 4 :
Divide $ 7 $ by $ \color{blue}{ 4 } $ and get the remainder

$$ 7 : 4 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 5 :
Divide $ 4 $ by $ \color{blue}{ 3 } $ and get the remainder

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

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.

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