Step 1 :
Divide $ 15670 $ by $ 10670 $ and get the remainder

$$ 15670 : 10670 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 10670 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 10670 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 670 } \text{ ) } $$

The remainder is still positive ($ 670 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 670 } $ and get the remainder

$$ 5000 : 670 = 7 \text{ ( remainder is } \color{blue}{ 310 } \text{ ) } $$

The remainder is still positive ($ 310 > 0 $), so we will continue with division.

Step 4 :
Divide $ 670 $ by $ \color{blue}{ 310 } $ and get the remainder

$$ 670 : 310 = 2 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

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

Step 5 :
Divide $ 310 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 310 : 50 = 6 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 6 :
Divide $ 50 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 50 : \color{blue}{ \boxed { 10 }} = 5 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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