Step 1 :
Divide $ 270 $ by $ 172 $ and get the remainder

$$ 270 : 172 = 1 \text{ ( remainder is } \color{blue}{ 98 } \text{ ) } $$

The remainder is positive ($ 98 > 0 $), so we will continue with division.

Step 2 :
Divide $ 172 $ by $ \color{blue}{ 98 } $ and get the remainder

$$ 172 : 98 = 1 \text{ ( remainder is } \color{blue}{ 74 } \text{ ) } $$

The remainder is still positive ($ 74 > 0 $), so we will continue with division.

Step 3 :
Divide $ 98 $ by $ \color{blue}{ 74 } $ and get the remainder

$$ 98 : 74 = 1 \text{ ( remainder is } \color{blue}{ 24 } \text{ ) } $$

The remainder is still positive ($ 24 > 0 $), so we will continue with division.

Step 4 :
Divide $ 74 $ by $ \color{blue}{ 24 } $ and get the remainder

$$ 74 : 24 = 3 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

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

Step 5 :
Divide $ 24 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 24 : \color{blue}{ \boxed { 2 }} = 12 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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