Step 1 :
Divide $ 2200765035 $ by $ 2000 $ and get the remainder

$$ 2200765035 : 2000 = 1100382 \text{ ( remainder is } \color{blue}{ 1035 } \text{ ) } $$

The remainder is positive ($ 1035 > 0 $), so we will continue with division.

Step 2 :
Divide $ 2000 $ by $ \color{blue}{ 1035 } $ and get the remainder

$$ 2000 : 1035 = 1 \text{ ( remainder is } \color{blue}{ 965 } \text{ ) } $$

The remainder is still positive ($ 965 > 0 $), so we will continue with division.

Step 3 :
Divide $ 1035 $ by $ \color{blue}{ 965 } $ and get the remainder

$$ 1035 : 965 = 1 \text{ ( remainder is } \color{blue}{ 70 } \text{ ) } $$

The remainder is still positive ($ 70 > 0 $), so we will continue with division.

Step 4 :
Divide $ 965 $ by $ \color{blue}{ 70 } $ and get the remainder

$$ 965 : 70 = 13 \text{ ( remainder is } \color{blue}{ 55 } \text{ ) } $$

The remainder is still positive ($ 55 > 0 $), so we will continue with division.

Step 5 :
Divide $ 70 $ by $ \color{blue}{ 55 } $ and get the remainder

$$ 70 : 55 = 1 \text{ ( remainder is } \color{blue}{ 15 } \text{ ) } $$

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

Step 6 :
Divide $ 55 $ by $ \color{blue}{ 15 } $ and get the remainder

$$ 55 : 15 = 3 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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.

GCD Calculator