Step 1 :
Divide $ 1003 $ by $ 821 $ and get the remainder

$$ 1003 : 821 = 1 \text{ ( remainder is } \color{blue}{ 182 } \text{ ) } $$

The remainder is positive ($ 182 > 0 $), so we will continue with division.

Step 2 :
Divide $ 821 $ by $ \color{blue}{ 182 } $ and get the remainder

$$ 821 : 182 = 4 \text{ ( remainder is } \color{blue}{ 93 } \text{ ) } $$

The remainder is still positive ($ 93 > 0 $), so we will continue with division.

Step 3 :
Divide $ 182 $ by $ \color{blue}{ 93 } $ and get the remainder

$$ 182 : 93 = 1 \text{ ( remainder is } \color{blue}{ 89 } \text{ ) } $$

The remainder is still positive ($ 89 > 0 $), so we will continue with division.

Step 4 :
Divide $ 93 $ by $ \color{blue}{ 89 } $ and get the remainder

$$ 93 : 89 = 1 \text{ ( remainder is } \color{blue}{ 4 } \text{ ) } $$

The remainder is still positive ($ 4 > 0 $), so we will continue with division.

Step 5 :
Divide $ 89 $ by $ \color{blue}{ 4 } $ and get the remainder

$$ 89 : 4 = 22 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 6 :
Divide $ 4 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 4 : \color{blue}{ \boxed { 1 }} = 4 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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