Step 1 :
Divide $ 7864 $ by $ 4148 $ and get the remainder

$$ 7864 : 4148 = 1 \text{ ( remainder is } \color{blue}{ 3716 } \text{ ) } $$

The remainder is positive ($ 3716 > 0 $), so we will continue with division.

Step 2 :
Divide $ 4148 $ by $ \color{blue}{ 3716 } $ and get the remainder

$$ 4148 : 3716 = 1 \text{ ( remainder is } \color{blue}{ 432 } \text{ ) } $$

The remainder is still positive ($ 432 > 0 $), so we will continue with division.

Step 3 :
Divide $ 3716 $ by $ \color{blue}{ 432 } $ and get the remainder

$$ 3716 : 432 = 8 \text{ ( remainder is } \color{blue}{ 260 } \text{ ) } $$

The remainder is still positive ($ 260 > 0 $), so we will continue with division.

Step 4 :
Divide $ 432 $ by $ \color{blue}{ 260 } $ and get the remainder

$$ 432 : 260 = 1 \text{ ( remainder is } \color{blue}{ 172 } \text{ ) } $$

The remainder is still positive ($ 172 > 0 $), so we will continue with division.

Step 5 :
Divide $ 260 $ by $ \color{blue}{ 172 } $ and get the remainder

$$ 260 : 172 = 1 \text{ ( remainder is } \color{blue}{ 88 } \text{ ) } $$

The remainder is still positive ($ 88 > 0 $), so we will continue with division.

Step 6 :
Divide $ 172 $ by $ \color{blue}{ 88 } $ and get the remainder

$$ 172 : 88 = 1 \text{ ( remainder is } \color{blue}{ 84 } \text{ ) } $$

The remainder is still positive ($ 84 > 0 $), so we will continue with division.

Step 7 :
Divide $ 88 $ by $ \color{blue}{ 84 } $ and get the remainder

$$ 88 : 84 = 1 \text{ ( remainder is } \color{blue}{ 4 } \text{ ) } $$

The remainder is still positive ($ 4 > 0 $), so we will continue with division.

Step 8 :
Divide $ 84 $ by $ \color{blue}{ 4 } $ and get the remainder

$$ 84 : \color{blue}{ \boxed { 4 }} = 21 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 4 }} $.

This solution can be visualized using a Venn diagram.

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

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