Step 1 :
Divide $ 345 $ by $ 217 $ and get the remainder

$$ 345 : 217 = 1 \text{ ( remainder is } \color{blue}{ 128 } \text{ ) } $$

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

Step 2 :
Divide $ 217 $ by $ \color{blue}{ 128 } $ and get the remainder

$$ 217 : 128 = 1 \text{ ( remainder is } \color{blue}{ 89 } \text{ ) } $$

The remainder is still positive ($ 89 > 0 $), so we will continue with division.

Step 3 :
Divide $ 128 $ by $ \color{blue}{ 89 } $ and get the remainder

$$ 128 : 89 = 1 \text{ ( remainder is } \color{blue}{ 39 } \text{ ) } $$

The remainder is still positive ($ 39 > 0 $), so we will continue with division.

Step 4 :
Divide $ 89 $ by $ \color{blue}{ 39 } $ and get the remainder

$$ 89 : 39 = 2 \text{ ( remainder is } \color{blue}{ 11 } \text{ ) } $$

The remainder is still positive ($ 11 > 0 $), so we will continue with division.

Step 5 :
Divide $ 39 $ by $ \color{blue}{ 11 } $ and get the remainder

$$ 39 : 11 = 3 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

The remainder is still positive ($ 6 > 0 $), so we will continue with division.

Step 6 :
Divide $ 11 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 11 : 6 = 1 \text{ ( remainder is } \color{blue}{ 5 } \text{ ) } $$

The remainder is still positive ($ 5 > 0 $), so we will continue with division.

Step 7 :
Divide $ 6 $ by $ \color{blue}{ 5 } $ and get the remainder

$$ 6 : 5 = 1 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

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

Step 8 :
Divide $ 5 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 5 : \color{blue}{ \boxed { 1 }} = 5 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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