Step 1 :
Divide $ 17650 $ by $ 12650 $ and get the remainder

$$ 17650 : 12650 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12650 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12650 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2650 } \text{ ) } $$

The remainder is still positive ($ 2650 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2650 } $ and get the remainder

$$ 5000 : 2650 = 1 \text{ ( remainder is } \color{blue}{ 2350 } \text{ ) } $$

The remainder is still positive ($ 2350 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2650 $ by $ \color{blue}{ 2350 } $ and get the remainder

$$ 2650 : 2350 = 1 \text{ ( remainder is } \color{blue}{ 300 } \text{ ) } $$

The remainder is still positive ($ 300 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2350 $ by $ \color{blue}{ 300 } $ and get the remainder

$$ 2350 : 300 = 7 \text{ ( remainder is } \color{blue}{ 250 } \text{ ) } $$

The remainder is still positive ($ 250 > 0 $), so we will continue with division.

Step 6 :
Divide $ 300 $ by $ \color{blue}{ 250 } $ and get the remainder

$$ 300 : 250 = 1 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

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

Step 7 :
Divide $ 250 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 250 : \color{blue}{ \boxed { 50 }} = 5 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 50 }} $.

This solution can be visualized using a Venn diagram.

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

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