Step 1 :
Divide $ 1000 $ by $ 625 $ and get the remainder

$$ 1000 : 625 = 1 \text{ ( remainder is } \color{blue}{ 375 } \text{ ) } $$

The remainder is positive ($ 375 > 0 $), so we will continue with division.

Step 2 :
Divide $ 625 $ by $ \color{blue}{ 375 } $ and get the remainder

$$ 625 : 375 = 1 \text{ ( remainder is } \color{blue}{ 250 } \text{ ) } $$

The remainder is still positive ($ 250 > 0 $), so we will continue with division.

Step 3 :
Divide $ 375 $ by $ \color{blue}{ 250 } $ and get the remainder

$$ 375 : 250 = 1 \text{ ( remainder is } \color{blue}{ 125 } \text{ ) } $$

The remainder is still positive ($ 125 > 0 $), so we will continue with division.

Step 4 :
Divide $ 250 $ by $ \color{blue}{ 125 } $ and get the remainder

$$ 250 : \color{blue}{ \boxed { 125 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 125 }} $.

This solution can be visualized using a Venn diagram.

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

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