query: Necessary and sufficient condition for $x \equiv u_1(mod \ m_1)$, $...$, $x \equiv u_n(mod \ m_n)$ to have a solution
positive_passages.docid: 22200
positive_passages.text: When a congruence system can be solved?
negative_passages.docid: 16548
negative_passages.text: Prove $(x+y) \text{ mod } n = ((x \text{ mod } n)+(y \text{ mod } n)) \text{ mod } n,\,$ and product analog
negative_passages.docid: 41890
negative_passages.text: How to get all solutions of $x^2 \equiv$ $a$ mod $p$?
negative_passages.docid: 77024
negative_passages.text: Find the number of solutions of $x^2=x\mod m$ for any integer $m$
negative_passages.docid: 29231
negative_passages.text: Proof of : $x^{n}\equiv x^{\varphi(m)+[n \bmod \varphi(m)]} \mod m$
negative_passages.docid: 132587
negative_passages.text: Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
negative_passages.docid: 38805
negative_passages.text: Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
negative_passages.docid: 7466
negative_passages.text: How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions
negative_passages.docid: 29204
negative_passages.text: $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
negative_passages.docid: 156034
negative_passages.text: sufficient condition for a polynomial to have roots in $[0,1]$
negative_passages.docid: 94965
negative_passages.text: Solve $x\equiv 1\pmod2$, $x\equiv 2\pmod3$, $x\equiv 3\pmod4$, $x\equiv 4\pmod5$, $x\equiv 5\pmod6$ and $x\equiv 0\pmod7$
negative_passages.docid: 7085
negative_passages.text: Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 71108
negative_passages.text: Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions?
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: 144851
negative_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
negative_passages.docid: 154933
negative_passages.text: Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$