Modular equation

There is only one solution to $x^2 \equiv 0 \mod p$ in the range specified by the problem:

$x = 0$

This can be seen as any square of any $x < p$ will not be divisible by $p$ . To prove this, note that the prime factorization of $x^2$ is the prime factorization of $x$ with the powers doubled, and that prime factorization will not include $p$ as $x < p$ . For $p$ to divide $x$ , $p$ would have to be included in the prime factorization. Therefore, the answer is $\boxed{0}$ .

We have to find sum of all x such that, $\large p |x^2, 0≤ x < p$ One can see, for every $p$ no such $positive$ $x$ exists. The only possibility is $\large p| 0$ So, $\large x = 0$ is only solution So answer is $\large\boxed0$