Let $a \in \mathbb{Z}$ and $[\mathrm{t}]$ be the greatest integer $\leq \mathrm{t}$. Then the number of points, where the function $f(x)=[a+13 \sin x], x \in(0, \pi)$ is not differentiable, is __________.

<p>Given that $ f(x) = [a + 13\sin(x)] $, where $[t]$ is the greatest integer function and $ x \in (0, \pi) $.</p> <p>The function $[t]$ is not differentiable wherever $ t $ is an integer because at these points, the function has a jump discontinuity.</p> <p>For $ f(x) $ to have a point of non-differentiability, the value inside the greatest integer function, i.e., $ a + 13\sin(x) $, should be an integer. </p> <p>So, we need to find the values of $ x $ in the interval $ (0, \pi) $ for which $ a + 13\sin(x) $ is an integer.</p> <p>Now, $ \sin(x) $ varies from 0 to 1 in the interval $ (0, \pi) $, so the maximum value of $ 13\sin(x) $ in this interval is 13.</p> <p>For each integer value of $ 13\sin(x) $ between 0 and 13, we&#39;ll have a corresponding value of $ x $. There will be two such values of $ x $ for each value of $ 13\sin(x) $ (because of the periodic and symmetric nature of sine function over the interval $ (0, \pi) $), except for the maximum value, 13, which will have only one corresponding value of $ x $ (namely $ x = \frac{\pi}{2} $).</p> <p>Thus, the total number of integer values of $ 13\sin(x) $ between 0 and 13 is 13. Excluding the maximum value, we have 12 integer values, each giving rise to two values of $ x $. Including the maximum value, which gives one value of $ x $, we have :</p> <p>$ 12 \times 2 + 1 = 25 $</p> <p>Thus, $ f(x) $ is not differentiable at 25 points in the interval $ (0, \pi) $.</p>