Step 1 :
Divide $ 627 $ by $ 168 $ and get the remainder

$$ 627 : 168 = 3 \text{ ( remainder is } \color{blue}{ 123 } \text{ ) } $$

The remainder is positive ($ 123 > 0 $), so we will continue with division.

Step 2 :
Divide $ 168 $ by $ \color{blue}{ 123 } $ and get the remainder

$$ 168 : 123 = 1 \text{ ( remainder is } \color{blue}{ 45 } \text{ ) } $$

The remainder is still positive ($ 45 > 0 $), so we will continue with division.

Step 3 :
Divide $ 123 $ by $ \color{blue}{ 45 } $ and get the remainder

$$ 123 : 45 = 2 \text{ ( remainder is } \color{blue}{ 33 } \text{ ) } $$

The remainder is still positive ($ 33 > 0 $), so we will continue with division.

Step 4 :
Divide $ 45 $ by $ \color{blue}{ 33 } $ and get the remainder

$$ 45 : 33 = 1 \text{ ( remainder is } \color{blue}{ 12 } \text{ ) } $$

The remainder is still positive ($ 12 > 0 $), so we will continue with division.

Step 5 :
Divide $ 33 $ by $ \color{blue}{ 12 } $ and get the remainder

$$ 33 : 12 = 2 \text{ ( remainder is } \color{blue}{ 9 } \text{ ) } $$

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

Step 6 :
Divide $ 12 $ by $ \color{blue}{ 9 } $ and get the remainder

$$ 12 : 9 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 7 :
Divide $ 9 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 9 : \color{blue}{ \boxed { 3 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

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