Recall that $\cos \theta = \cos (360^\circ - \theta)$ for all angles $\theta$.
Therefore, $\cos n^\circ = \cos 259^\circ$ if and only if $n^\circ = 259^\circ$ or $n^\circ = 360^\circ - 259^\circ = 101^\circ$.
Since $0 \le n \le 180$, the only possible value for $n$ is $\boxed{101}$.
The answer is: 101
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 259^\circ.$

### Answer:
