Step 1 :
Divide $ 17770 $ by $ 12770 $ and get the remainder

$$ 17770 : 12770 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

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

Step 2 :
Divide $ 12770 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12770 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2770 } \text{ ) } $$

The remainder is still positive ($ 2770 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2770 } $ and get the remainder

$$ 5000 : 2770 = 1 \text{ ( remainder is } \color{blue}{ 2230 } \text{ ) } $$

The remainder is still positive ($ 2230 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2770 $ by $ \color{blue}{ 2230 } $ and get the remainder

$$ 2770 : 2230 = 1 \text{ ( remainder is } \color{blue}{ 540 } \text{ ) } $$

The remainder is still positive ($ 540 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2230 $ by $ \color{blue}{ 540 } $ and get the remainder

$$ 2230 : 540 = 4 \text{ ( remainder is } \color{blue}{ 70 } \text{ ) } $$

The remainder is still positive ($ 70 > 0 $), so we will continue with division.

Step 6 :
Divide $ 540 $ by $ \color{blue}{ 70 } $ and get the remainder

$$ 540 : 70 = 7 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

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

Step 7 :
Divide $ 70 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 70 : 50 = 1 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

The remainder is still positive ($ 20 > 0 $), so we will continue with division.

Step 8 :
Divide $ 50 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 50 : 20 = 2 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

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

Step 9 :
Divide $ 20 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 20 : \color{blue}{ \boxed { 10 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

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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