Step 1 :
Divide $ 380 $ by $ 275 $ and get the remainder

$$ 380 : 275 = 1 \text{ ( remainder is } \color{blue}{ 105 } \text{ ) } $$

The remainder is positive ($ 105 > 0 $), so we will continue with division.

Step 2 :
Divide $ 275 $ by $ \color{blue}{ 105 } $ and get the remainder

$$ 275 : 105 = 2 \text{ ( remainder is } \color{blue}{ 65 } \text{ ) } $$

The remainder is still positive ($ 65 > 0 $), so we will continue with division.

Step 3 :
Divide $ 105 $ by $ \color{blue}{ 65 } $ and get the remainder

$$ 105 : 65 = 1 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

The remainder is still positive ($ 40 > 0 $), so we will continue with division.

Step 4 :
Divide $ 65 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 65 : 40 = 1 \text{ ( remainder is } \color{blue}{ 25 } \text{ ) } $$

The remainder is still positive ($ 25 > 0 $), so we will continue with division.

Step 5 :
Divide $ 40 $ by $ \color{blue}{ 25 } $ and get the remainder

$$ 40 : 25 = 1 \text{ ( remainder is } \color{blue}{ 15 } \text{ ) } $$

The remainder is still positive ($ 15 > 0 $), so we will continue with division.

Step 6 :
Divide $ 25 $ by $ \color{blue}{ 15 } $ and get the remainder

$$ 25 : 15 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

The remainder is still positive ($ 10 > 0 $), so we will continue with division.

Step 7 :
Divide $ 15 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 15 : 10 = 1 \text{ ( remainder is } \color{blue}{ 5 } \text{ ) } $$

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

Step 8 :
Divide $ 10 $ by $ \color{blue}{ 5 } $ and get the remainder

$$ 10 : \color{blue}{ \boxed { 5 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 5 }} $.

This solution can be visualized using a Venn diagram.

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

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