Step 1 :
Divide $ 623 $ by $ 59 $ and get the remainder

$$ 623 : 59 = 10 \text{ ( remainder is } \color{blue}{ 33 } \text{ ) } $$

The remainder is positive ($ 33 > 0 $), so we will continue with division.

Step 2 :
Divide $ 59 $ by $ \color{blue}{ 33 } $ and get the remainder

$$ 59 : 33 = 1 \text{ ( remainder is } \color{blue}{ 26 } \text{ ) } $$

The remainder is still positive ($ 26 > 0 $), so we will continue with division.

Step 3 :
Divide $ 33 $ by $ \color{blue}{ 26 } $ and get the remainder

$$ 33 : 26 = 1 \text{ ( remainder is } \color{blue}{ 7 } \text{ ) } $$

The remainder is still positive ($ 7 > 0 $), so we will continue with division.

Step 4 :
Divide $ 26 $ by $ \color{blue}{ 7 } $ and get the remainder

$$ 26 : 7 = 3 \text{ ( remainder is } \color{blue}{ 5 } \text{ ) } $$

The remainder is still positive ($ 5 > 0 $), so we will continue with division.

Step 5 :
Divide $ 7 $ by $ \color{blue}{ 5 } $ and get the remainder

$$ 7 : 5 = 1 \text{ ( remainder is } \color{blue}{ 2 } \text{ ) } $$

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

Step 6 :
Divide $ 5 $ by $ \color{blue}{ 2 } $ and get the remainder

$$ 5 : 2 = 2 \text{ ( remainder is } \color{blue}{ 1 } \text{ ) } $$

The remainder is still positive ($ 1 > 0 $), so we will continue with division.

Step 7 :
Divide $ 2 $ by $ \color{blue}{ 1 } $ and get the remainder

$$ 2 : \color{blue}{ \boxed { 1 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 1 }} $.

This solution can be visualized using a Venn diagram.

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

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