Step 1 :
Divide $ 4601 $ by $ 2021 $ and get the remainder

$$ 4601 : 2021 = 2 \text{ ( remainder is } \color{blue}{ 559 } \text{ ) } $$

The remainder is positive ($ 559 > 0 $), so we will continue with division.

Step 2 :
Divide $ 2021 $ by $ \color{blue}{ 559 } $ and get the remainder

$$ 2021 : 559 = 3 \text{ ( remainder is } \color{blue}{ 344 } \text{ ) } $$

The remainder is still positive ($ 344 > 0 $), so we will continue with division.

Step 3 :
Divide $ 559 $ by $ \color{blue}{ 344 } $ and get the remainder

$$ 559 : 344 = 1 \text{ ( remainder is } \color{blue}{ 215 } \text{ ) } $$

The remainder is still positive ($ 215 > 0 $), so we will continue with division.

Step 4 :
Divide $ 344 $ by $ \color{blue}{ 215 } $ and get the remainder

$$ 344 : 215 = 1 \text{ ( remainder is } \color{blue}{ 129 } \text{ ) } $$

The remainder is still positive ($ 129 > 0 $), so we will continue with division.

Step 5 :
Divide $ 215 $ by $ \color{blue}{ 129 } $ and get the remainder

$$ 215 : 129 = 1 \text{ ( remainder is } \color{blue}{ 86 } \text{ ) } $$

The remainder is still positive ($ 86 > 0 $), so we will continue with division.

Step 6 :
Divide $ 129 $ by $ \color{blue}{ 86 } $ and get the remainder

$$ 129 : 86 = 1 \text{ ( remainder is } \color{blue}{ 43 } \text{ ) } $$

The remainder is still positive ($ 43 > 0 $), so we will continue with division.

Step 7 :
Divide $ 86 $ by $ \color{blue}{ 43 } $ and get the remainder

$$ 86 : \color{blue}{ \boxed { 43 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 43 }} $.

This solution can be visualized using a Venn diagram.

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

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