Step 1 : Find prime factorization of each number.

$$\begin{aligned}8125 =& 5\cdot5\cdot5\cdot5\cdot13\\[8pt]10000 =& 2\cdot2\cdot2\cdot2\cdot5\cdot5\cdot5\cdot5\\[8pt]\end{aligned}$$

(view steps on how to factor 8125 and 10000. )

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

$$\begin{aligned}8125 =& \color{blue}{\boxed{5}}\cdot\color{red}{\boxed{5}}\cdot\color{Fuchsia}{\boxed{5}}\cdot\color{Orange}{\boxed{5}}\cdot13\\[8pt]10000 =& 2\cdot2\cdot2\cdot2\cdot\color{blue}{\boxed{5}}\cdot\color{red}{\boxed{5}}\cdot\color{Fuchsia}{\boxed{5}}\cdot\color{Orange}{\boxed{5}}\\[8pt]\end{aligned}$$

Step 3 : Multiply the boxed numbers together:

$$ GCD = 5\cdot5\cdot5\cdot5 = 625 $$

This solution can be visualized using a Venn diagram.

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

GCD Calculator