Step 1 :
Divide $ 822 $ by $ 436 $ and get the remainder

$$ 822 : 436 = 1 \text{ ( remainder is } \color{blue}{ 386 } \text{ ) } $$

The remainder is positive ($ 386 > 0 $), so we will continue with division.

Step 2 :
Divide $ 436 $ by $ \color{blue}{ 386 } $ and get the remainder

$$ 436 : 386 = 1 \text{ ( remainder is } \color{blue}{ 50 } \text{ ) } $$

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

Step 3 :
Divide $ 386 $ by $ \color{blue}{ 50 } $ and get the remainder

$$ 386 : 50 = 7 \text{ ( remainder is } \color{blue}{ 36 } \text{ ) } $$

The remainder is still positive ($ 36 > 0 $), so we will continue with division.

Step 4 :
Divide $ 50 $ by $ \color{blue}{ 36 } $ and get the remainder

$$ 50 : 36 = 1 \text{ ( remainder is } \color{blue}{ 14 } \text{ ) } $$

The remainder is still positive ($ 14 > 0 $), so we will continue with division.

Step 5 :
Divide $ 36 $ by $ \color{blue}{ 14 } $ and get the remainder

$$ 36 : 14 = 2 \text{ ( remainder is } \color{blue}{ 8 } \text{ ) } $$

The remainder is still positive ($ 8 > 0 $), so we will continue with division.

Step 6 :
Divide $ 14 $ by $ \color{blue}{ 8 } $ and get the remainder

$$ 14 : 8 = 1 \text{ ( remainder is } \color{blue}{ 6 } \text{ ) } $$

The remainder is still positive ($ 6 > 0 $), so we will continue with division.

Step 7 :
Divide $ 8 $ by $ \color{blue}{ 6 } $ and get the remainder

$$ 8 : 6 = 1 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

The remainder is still positive ($ 2 > 0 $), so we will continue with division.

Step 8 :
Divide $ 6 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 6 : \color{blue}{ \boxed { 2 }} = 3 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 2 }} $.

This solution can be visualized using a Venn diagram.

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

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