Step 1 :
Divide $ 1001 $ by $ 128 $ and get the remainder

$$ 1001 : 128 = 7 \text{ ( remainder is } \color{blue}{ 105 } \text{ ) } $$

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

Step 2 :
Divide $ 128 $ by $ \color{blue}{ 105 } $ and get the remainder

$$ 128 : 105 = 1 \text{ ( remainder is } \color{blue}{ 23 } \text{ ) } $$

The remainder is still positive ($ 23 > 0 $), so we will continue with division.

Step 3 :
Divide $ 105 $ by $ \color{blue}{ 23 } $ and get the remainder

$$ 105 : 23 = 4 \text{ ( remainder is } \color{blue}{ 13 } \text{ ) } $$

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

Step 4 :
Divide $ 23 $ by $ \color{blue}{ 13 } $ and get the remainder

$$ 23 : 13 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 5 :
Divide $ 13 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 13 : 10 = 1 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

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

Step 6 :
Divide $ 10 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 10 : 3 = 3 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 7 :
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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