Step 1 :
Divide $ 17670 $ by $ 12670 $ and get the remainder

$$ 17670 : 12670 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12670 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12670 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2670 } \text{ ) } $$

The remainder is still positive ($ 2670 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2670 } $ and get the remainder

$$ 5000 : 2670 = 1 \text{ ( remainder is } \color{blue}{ 2330 } \text{ ) } $$

The remainder is still positive ($ 2330 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2670 $ by $ \color{blue}{ 2330 } $ and get the remainder

$$ 2670 : 2330 = 1 \text{ ( remainder is } \color{blue}{ 340 } \text{ ) } $$

The remainder is still positive ($ 340 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2330 $ by $ \color{blue}{ 340 } $ and get the remainder

$$ 2330 : 340 = 6 \text{ ( remainder is } \color{blue}{ 290 } \text{ ) } $$

The remainder is still positive ($ 290 > 0 $), so we will continue with division.

Step 6 :
Divide $ 340 $ by $ \color{blue}{ 290 } $ and get the remainder

$$ 340 : 290 = 1 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

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

Step 7 :
Divide $ 290 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 290 : 50 = 5 \text{ ( remainder is } \color{blue}{ 40 } \text{ ) } $$

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

Step 8 :
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 9 :
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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