Step 1 : Find prime factorization of each number.

$$\begin{aligned}67 =& 67\\[8pt]653 =& 653\\[8pt]577 =& 577\\[8pt]567 =& 3\cdot3\cdot3\cdot3\cdot7\\[8pt]441 =& 3\cdot3\cdot7\cdot7\\[8pt]417 =& 3\cdot139\\[8pt]457 =& 457\\[8pt]378 =& 2\cdot3\cdot3\cdot3\cdot7\\[8pt]306 =& 2\cdot3\cdot3\cdot17\\[8pt]558 =& 2\cdot3\cdot3\cdot31\\[8pt]387 =& 3\cdot3\cdot43\\[8pt]\end{aligned}$$

(view steps on how to factor 67, 653, 577, 567, 441, 417, 457, 378, 306, 558 and 387. )

Step 2 : Put a box around factors that are common for all numbers:

$$\begin{aligned}67 =& 67\\[8pt]653 =& 653\\[8pt]577 =& 577\\[8pt]567 =& 3\cdot3\cdot3\cdot3\cdot7\\[8pt]441 =& 3\cdot3\cdot7\cdot7\\[8pt]417 =& 3\cdot139\\[8pt]457 =& 457\\[8pt]378 =& 2\cdot3\cdot3\cdot3\cdot7\\[8pt]306 =& 2\cdot3\cdot3\cdot17\\[8pt]558 =& 2\cdot3\cdot3\cdot31\\[8pt]387 =& 3\cdot3\cdot43\\[8pt]\end{aligned}$$

Note that in this example numbers do not have any common factors.

Step 3 : Multiply the boxed numbers together:

Since there is no boxed numbers we conclude that $~\color{blue}{ \text{GCD = 1} } $.

GCD Calculator