Step 1 :
Divide $ 17110 $ by $ 12110 $ and get the remainder

$$ 17110 : 12110 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

The remainder is positive ($ 5000 > 0 $), so we will continue with division.

Step 2 :
Divide $ 12110 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12110 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2110 } \text{ ) } $$

The remainder is still positive ($ 2110 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2110 } $ and get the remainder

$$ 5000 : 2110 = 2 \text{ ( remainder is } \color{blue}{ 780 } \text{ ) } $$

The remainder is still positive ($ 780 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2110 $ by $ \color{blue}{ 780 } $ and get the remainder

$$ 2110 : 780 = 2 \text{ ( remainder is } \color{blue}{ 550 } \text{ ) } $$

The remainder is still positive ($ 550 > 0 $), so we will continue with division.

Step 5 :
Divide $ 780 $ by $ \color{blue}{ 550 } $ and get the remainder

$$ 780 : 550 = 1 \text{ ( remainder is } \color{blue}{ 230 } \text{ ) } $$

The remainder is still positive ($ 230 > 0 $), so we will continue with division.

Step 6 :
Divide $ 550 $ by $ \color{blue}{ 230 } $ and get the remainder

$$ 550 : 230 = 2 \text{ ( remainder is } \color{blue}{ 90 } \text{ ) } $$

The remainder is still positive ($ 90 > 0 $), so we will continue with division.

Step 7 :
Divide $ 230 $ by $ \color{blue}{ 90 } $ and get the remainder

$$ 230 : 90 = 2 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

The remainder is still positive ($ 50 > 0 $), so we will continue with division.

Step 8 :
Divide $ 90 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 90 : 50 = 1 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

The remainder is still positive ($ 40 > 0 $), so we will continue with division.

Step 9 :
Divide $ 50 $ by $ \color{blue}{ 40 } $ and get the remainder

$$ 50 : 40 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 10 :
Divide $ 40 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 40 : \color{blue}{ \boxed { 10 }} = 4 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 10 }} $.

This solution can be visualized using a Venn diagram.

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

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