10. (10 points) Given 100 distinct prime numbers $p_{1}, p_{2}, \cdots, p_{100}$, let $N=p_{1}^{2}+p_{1}^{2}+\cdots+p_{100}^{2}$, what is the remainder when $N$ is divided by 3?
【Analysis】Except for 3, since the remainder of a prime number divided by 3 is 1 or 2, the remainder of the square of a prime number divided by 3 can only be 1 (the square of 2 divided by 3 leaves a remainder of 1). Then, we discuss whether it contains the prime number 3.
【Solution】Solution: (1) If these primes do not contain the prime number 3, then the remainder of the square of this number divided by 3 is 1, so the remainder of $N$ divided by 3 is the same as the remainder of 100 divided by 3, which is 1;
(2) If it contains 3, then the remaining 99 leave a remainder of 0.
The square of 3 divided by 3 leaves a remainder of 0.
Thus, the remainder of $N$ divided by 3 is 0.
Answer: The remainder of $N$ divided by 3 is 0 or 1.