Step 1 : Find prime factorization of each number.

$$\begin{aligned}15210 =& 2\cdot3\cdot3\cdot5\cdot13\cdot13\\[8pt]10210 =& 2\cdot5\cdot1021\\[8pt]\end{aligned}$$

(view steps on how to factor 15210 and 10210. )

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

$$\begin{aligned}15210 =& \color{blue}{\boxed{2}}\cdot3\cdot3\cdot\color{red}{\boxed{5}}\cdot13\cdot13\\[8pt]10210 =& \color{blue}{\boxed{2}}\cdot\color{red}{\boxed{5}}\cdot1021\\[8pt]\end{aligned}$$

Step 3 : Multiply the boxed numbers together:

$$ GCD = 2\cdot5 = 10 $$

This solution can be visualized using a Venn diagram.

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

GCD Calculator