Step-by-step Solution:

Given: \[ \sum_{i=1}^n x_i = 80 \quad \text{and} \quad \sum_{i=1}^n x_i^2 = 400 \] Using the RMS ≥ AM inequality: \[ \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}} \geq \frac{x_1 + \cdots + x_n}{n} \] Substitute the given sums: \[ \sqrt{\frac{400}{n}} \geq \frac{80}{n} \] Square both sides to eliminate the square root: \[ \frac{400}{n} \geq \left(\frac{80}{n}\right)^2 \quad \Rightarrow \quad \frac{400}{n} \geq \frac{6400}{n^2} \] Multiply through by \( n^2 \) (since \( n > 0 \)): \[ 400n \geq 6400 \quad \Rightarrow \quad n \geq 16 \]