query: $a\equiv b\pmod n$ iff $a \bmod m = b \bmod m$
positive_passages.docid: 30725
positive_passages.text: Why are naturals < modulus equal if congruent? [uniqueness of remainder]
negative_passages.docid: 10105
negative_passages.text: Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
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: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
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: 19071
negative_passages.text: solution set for congruence $x^2 \equiv 1 \mod m$
negative_passages.docid: 97196
negative_passages.text: $k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
negative_passages.docid: 21737
negative_passages.text: If $A|B$ and $B|A$ then prove $A=\pm B$
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: 42780
negative_passages.text: Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
negative_passages.docid: 81346
negative_passages.text: Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.
negative_passages.docid: 136804
negative_passages.text: Computing $\mathrm{Hom}(\mathbb Z_n,\mathbb Z_m)$ as $\mathbb Z$-module
negative_passages.docid: 41673
negative_passages.text: Solve congruence system $\ x\equiv m_i-1 \pmod{m_i}\,$ for $\,i = 1,\ldots, k$
negative_passages.docid: 51153
negative_passages.text: Number of solutions to $x^2\equiv a\pmod m$
negative_passages.docid: 21250
negative_passages.text: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
negative_passages.docid: 4972
negative_passages.text: Mod of numbers with large exponents [modular order reduction]
negative_passages.docid: 96237
negative_passages.text: $1^2 + 2^2 + 3^2 + \cdots + (n-1)^2 \equiv 0$ mod n iff $n \equiv \pm 1$ mod 6