The GCD of given numbers is 27.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}513 =& 3\cdot3\cdot3\cdot19\\[8pt]783 =& 3\cdot3\cdot3\cdot29\\[8pt]1107 =& 3\cdot3\cdot3\cdot41\\[8pt]\end{aligned}$$(view steps on how to factor 513, 783 and 1107. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}513 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{3}}\cdot\color{Fuchsia}{\boxed{3}}\cdot19\\[8pt]783 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{3}}\cdot\color{Fuchsia}{\boxed{3}}\cdot29\\[8pt]1107 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{3}}\cdot\color{Fuchsia}{\boxed{3}}\cdot41\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 3\cdot3\cdot3 = 27 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.