Step 1 :
Divide $ 1214 $ by $ 460 $ and get the remainder

$$ 1214 : 460 = 2 \text{ ( remainder is } \color{blue}{ 294 } \text{ ) } $$

The remainder is positive ($ 294 > 0 $), so we will continue with division.

Step 2 :
Divide $ 460 $ by $ \color{blue}{ 294 } $ and get the remainder

$$ 460 : 294 = 1 \text{ ( remainder is } \color{blue}{ 166 } \text{ ) } $$

The remainder is still positive ($ 166 > 0 $), so we will continue with division.

Step 3 :
Divide $ 294 $ by $ \color{blue}{ 166 } $ and get the remainder

$$ 294 : 166 = 1 \text{ ( remainder is } \color{blue}{ 128 } \text{ ) } $$

The remainder is still positive ($ 128 > 0 $), so we will continue with division.

Step 4 :
Divide $ 166 $ by $ \color{blue}{ 128 } $ and get the remainder

$$ 166 : 128 = 1 \text{ ( remainder is } \color{blue}{ 38 } \text{ ) } $$

The remainder is still positive ($ 38 > 0 $), so we will continue with division.

Step 5 :
Divide $ 128 $ by $ \color{blue}{ 38 } $ and get the remainder

$$ 128 : 38 = 3 \text{ ( remainder is } \color{blue}{ 14 } \text{ ) } $$

The remainder is still positive ($ 14 > 0 $), so we will continue with division.

Step 6 :
Divide $ 38 $ by $ \color{blue}{ 14 } $ and get the remainder

$$ 38 : 14 = 2 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

The remainder is still positive ($ 10 > 0 $), so we will continue with division.

Step 7 :
Divide $ 14 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 14 : 10 = 1 \text{ ( remainder is } \color{blue}{ 4 } \text{ ) } $$

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

Step 8 :
Divide $ 10 $ by $ \color{blue}{ 4 } $ and get the remainder

$$ 10 : 4 = 2 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

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

Step 9 :
Divide $ 4 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 4 : \color{blue}{ \boxed { 2 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 2 }} $.

This solution can be visualized using a Venn diagram.

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

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