Find the number of five-digit positive integers, $n$, that satisfy the following conditions:


(a) the number $n$ is divisible by $5,$


(b) the first and last digits of $n$ are equal, and


(c) the sum of the digits of $n$ is divisible by $5.$
The number takes a form of $5\text{x,y,z}5$, in which $5|(x+y+z)$. Let $x$ and $y$ be arbitrary digits. For each pair of $x,y$, there are exactly two values of $z$ that satisfy the condition of $5|(x+y+z)$. Therefore, the answer is $10\times10\times2=\boxed{200}$