We can prime factorize $144$ as $2^4 \cdot 3^2$. 
So, $2^x = 2^4$ and $3^y = 3^2$. 
Therefore, $x = 4$ and $y = 2$. 
Substituting these values into the expression, we have $\left(\frac15\right)^{2 - 4} = \left(\frac15\right)^{-2} = \boxed{25}$. 
The answer is: 25
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $2^x$ be the greatest power of $2$ that is a factor of $144$, and let $3^y$ be the greatest power of $3$ that is a factor of $144$. Evaluate the following expression: $$\left(\frac15\right)^{y - x}$$

### Answer:
