query: Consider the congruence $x^2 − 1 \equiv 0 \pmod{8}$. How many solutions does it have with $0 \le x < 8$?
positive_passages.docid: 27063
positive_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 45882
negative_passages.text: Solving a congruence without Fermat's little theorem
negative_passages.docid: 14809
negative_passages.text: Existence & uniqueness of solutions of linear congruences $\ cx\equiv b\pmod m$
negative_passages.docid: 33314
negative_passages.text: How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
negative_passages.docid: 24837
negative_passages.text: $x^2+1=0$ uncountable many solutions
negative_passages.docid: 133925
negative_passages.text: Find All Solutions to System of Congruence
negative_passages.docid: 94407
negative_passages.text: Number of solutions for $x[1] + x[2] + \ldots + x[n] =k$
negative_passages.docid: 94917
negative_passages.text: Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
negative_passages.docid: 126725
negative_passages.text: If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
negative_passages.docid: 131941
negative_passages.text: Solve quadratic congruences.
negative_passages.docid: 69858
negative_passages.text: Congruence Equation $3n^3+12n^2+13n+2\equiv0,\pmod{2\times3\times5}$
negative_passages.docid: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 22029
negative_passages.text: Solving modular arithmetic questions
negative_passages.docid: 135465
negative_passages.text: Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
negative_passages.docid: 56803
negative_passages.text: Finding solutions to congruence equations
negative_passages.docid: 140763
negative_passages.text: How to prove that $\sum_{n=1}^{2012} \frac{1}{x_n} = 1 $ has finitely many solutions for positive integer $x_i$?