The GCD of given numbers is 1008.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}16128 =& 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot7\\[8pt]27216 =& 2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot3\cdot3\cdot7\\[8pt]25200 =& 2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot5\cdot5\cdot7\\[8pt]\end{aligned}$$(view steps on how to factor 16128, 27216 and 25200. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}16128 =& \color{blue}{\boxed{2}}\cdot\color{red}{\boxed{2}}\cdot\color{Fuchsia}{\boxed{2}}\cdot\color{Orange}{\boxed{2}}\cdot2\cdot2\cdot2\cdot2\cdot\color{Purple}{\boxed{3}}\cdot\color{blue}{\boxed{3}}\cdot\color{red}{\boxed{7}}\\[8pt]27216 =& \color{blue}{\boxed{2}}\cdot\color{red}{\boxed{2}}\cdot\color{Fuchsia}{\boxed{2}}\cdot\color{Orange}{\boxed{2}}\cdot\color{Purple}{\boxed{3}}\cdot\color{blue}{\boxed{3}}\cdot3\cdot3\cdot3\cdot\color{red}{\boxed{7}}\\[8pt]25200 =& \color{blue}{\boxed{2}}\cdot\color{red}{\boxed{2}}\cdot\color{Fuchsia}{\boxed{2}}\cdot\color{Orange}{\boxed{2}}\cdot\color{Purple}{\boxed{3}}\cdot\color{blue}{\boxed{3}}\cdot5\cdot5\cdot\color{red}{\boxed{7}}\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot7 = 1008 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.