Step 1 : Find prime factorization of each number.

$$\begin{aligned}293 =& 293\\[8pt]7298 =& 2\cdot41\cdot89\\[8pt]19229 =& 7\cdot41\cdot67\\[8pt]\end{aligned}$$

(view steps on how to factor 293, 7298 and 19229. )

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

$$\begin{aligned}293 =& 293\\[8pt]7298 =& 2\cdot41\cdot89\\[8pt]19229 =& 7\cdot41\cdot67\\[8pt]\end{aligned}$$

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} } $.

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

GCD Calculator