Step 1 :
Divide $ 12211 $ by $ 10422 $ and get the remainder

$$ 12211 : 10422 = 1 \text{ ( remainder is } \color{blue}{ 1789 } \text{ ) } $$

The remainder is positive ($ 1789 > 0 $), so we will continue with division.

Step 2 :
Divide $ 10422 $ by $ \color{blue}{ 1789 } $ and get the remainder

$$ 10422 : 1789 = 5 \text{ ( remainder is } \color{blue}{ 1477 } \text{ ) } $$

The remainder is still positive ($ 1477 > 0 $), so we will continue with division.

Step 3 :
Divide $ 1789 $ by $ \color{blue}{ 1477 } $ and get the remainder

$$ 1789 : 1477 = 1 \text{ ( remainder is } \color{blue}{ 312 } \text{ ) } $$

The remainder is still positive ($ 312 > 0 $), so we will continue with division.

Step 4 :
Divide $ 1477 $ by $ \color{blue}{ 312 } $ and get the remainder

$$ 1477 : 312 = 4 \text{ ( remainder is } \color{blue}{ 229 } \text{ ) } $$

The remainder is still positive ($ 229 > 0 $), so we will continue with division.

Step 5 :
Divide $ 312 $ by $ \color{blue}{ 229 } $ and get the remainder

$$ 312 : 229 = 1 \text{ ( remainder is } \color{blue}{ 83 } \text{ ) } $$

The remainder is still positive ($ 83 > 0 $), so we will continue with division.

Step 6 :
Divide $ 229 $ by $ \color{blue}{ 83 } $ and get the remainder

$$ 229 : 83 = 2 \text{ ( remainder is } \color{blue}{ 63 } \text{ ) } $$

The remainder is still positive ($ 63 > 0 $), so we will continue with division.

Step 7 :
Divide $ 83 $ by $ \color{blue}{ 63 } $ and get the remainder

$$ 83 : 63 = 1 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

The remainder is still positive ($ 20 > 0 $), so we will continue with division.

Step 8 :
Divide $ 63 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 63 : 20 = 3 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

The remainder is still positive ($ 3 > 0 $), so we will continue with division.

Step 9 :
Divide $ 20 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 20 : 3 = 6 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

The remainder is still positive ($ 2 > 0 $), so we will continue with division.

Step 10 :
Divide $ 3 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 3 : 2 = 1 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 11 :
Divide $ 2 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 2 : \color{blue}{ \boxed { 1 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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