Step 1 :
Divide $ 1274 $ by $ 460 $ and get the remainder

$$ 1274 : 460 = 2 \text{ ( remainder is } \color{blue}{ 354 } \text{ ) } $$

The remainder is positive ($ 354 > 0 $), so we will continue with division.

Step 2 :
Divide $ 460 $ by $ \color{blue}{ 354 } $ and get the remainder

$$ 460 : 354 = 1 \text{ ( remainder is } \color{blue}{ 106 } \text{ ) } $$

The remainder is still positive ($ 106 > 0 $), so we will continue with division.

Step 3 :
Divide $ 354 $ by $ \color{blue}{ 106 } $ and get the remainder

$$ 354 : 106 = 3 \text{ ( remainder is } \color{blue}{ 36 } \text{ ) } $$

The remainder is still positive ($ 36 > 0 $), so we will continue with division.

Step 4 :
Divide $ 106 $ by $ \color{blue}{ 36 } $ and get the remainder

$$ 106 : 36 = 2 \text{ ( remainder is } \color{blue}{ 34 } \text{ ) } $$

The remainder is still positive ($ 34 > 0 $), so we will continue with division.

Step 5 :
Divide $ 36 $ by $ \color{blue}{ 34 } $ and get the remainder

$$ 36 : 34 = 1 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

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

Step 6 :
Divide $ 34 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 34 : \color{blue}{ \boxed { 2 }} = 17 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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.

GCD Calculator