Step 1 :
Divide $ 446 $ by $ 269 $ and get the remainder

$$ 446 : 269 = 1 \text{ ( remainder is } \color{blue}{ 177 } \text{ ) } $$

The remainder is positive ($ 177 > 0 $), so we will continue with division.

Step 2 :
Divide $ 269 $ by $ \color{blue}{ 177 } $ and get the remainder

$$ 269 : 177 = 1 \text{ ( remainder is } \color{blue}{ 92 } \text{ ) } $$

The remainder is still positive ($ 92 > 0 $), so we will continue with division.

Step 3 :
Divide $ 177 $ by $ \color{blue}{ 92 } $ and get the remainder

$$ 177 : 92 = 1 \text{ ( remainder is } \color{blue}{ 85 } \text{ ) } $$

The remainder is still positive ($ 85 > 0 $), so we will continue with division.

Step 4 :
Divide $ 92 $ by $ \color{blue}{ 85 } $ and get the remainder

$$ 92 : 85 = 1 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

The remainder is still positive ($ 7 > 0 $), so we will continue with division.

Step 5 :
Divide $ 85 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 85 : 7 = 12 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 6 :
Divide $ 7 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 7 : \color{blue}{ \boxed { 1 }} = 7 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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