Step 1 :
Divide $ 876 $ by $ 135 $ and get the remainder

$$ 876 : 135 = 6 \text{ ( remainder is } \color{blue}{ 66 } \text{ ) } $$

The remainder is positive ($ 66 > 0 $), so we will continue with division.

Step 2 :
Divide $ 135 $ by $ \color{blue}{ 66 } $ and get the remainder

$$ 135 : 66 = 2 \text{ ( remainder is } \color{blue}{ 3 } \text{ ) } $$

The remainder is still positive ($ 3 > 0 $), so we will continue with division.

Step 3 :
Divide $ 66 $ by $ \color{blue}{ 3 } $ and get the remainder

$$ 66 : \color{blue}{ \boxed { 3 }} = 22 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.

This solution can be visualized using a Venn diagram.

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

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