The GCD of given numbers is 22.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}35904 =& 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot3\cdot11\cdot17\\[8pt]5170 =& 2\cdot5\cdot11\cdot47\\[8pt]\end{aligned}$$(view steps on how to factor 35904 and 5170. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}35904 =& \color{blue}{\boxed{2}}\cdot2\cdot2\cdot2\cdot2\cdot2\cdot3\cdot\color{red}{\boxed{11}}\cdot17\\[8pt]5170 =& \color{blue}{\boxed{2}}\cdot5\cdot\color{red}{\boxed{11}}\cdot47\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 2\cdot11 = 22 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.