The GCD of given numbers is 21.
Step 1 : Find prime factorization of each number.
$$\begin{aligned}85743 =& 3\cdot3\cdot7\cdot1361\\[8pt]256767 =& 3\cdot7\cdot12227\\[8pt]\end{aligned}$$(view steps on how to factor 85743 and 256767. )
Step 2 : Put a box around factors that are common for all numbers:
$$\begin{aligned}85743 =& \color{blue}{\boxed{3}}\cdot3\cdot\color{red}{\boxed{7}}\cdot1361\\[8pt]256767 =& \color{blue}{\boxed{3}}\cdot\color{red}{\boxed{7}}\cdot12227\\[8pt]\end{aligned}$$Step 3 : Multiply the boxed numbers together:
$$ GCD = 3\cdot7 = 21 $$This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.