query: If $ a,m,x \in \Bbb{N}$ and $m\geq 2$ with $x | m $ and $x | a\iff x | (a\bmod m) $
positive_passages.docid: 157155
positive_passages.text: How is HCF (c, d) = HCF (d, r), where c = dq +r (by Euclid's division lemma)?
negative_passages.docid: 139659
negative_passages.text: Find all $n$ with $\!\bmod n\!:\, a\,$ invertible $\Rightarrow a^{-1} \equiv a,\,$ i.e. $a^2\equiv 1$
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 98496
negative_passages.text: Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
negative_passages.docid: 135986
negative_passages.text: Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
negative_passages.docid: 107142
negative_passages.text: If $(x_n) \to x$ then $(\sqrt[n]{x_1x_2\cdots x_n}) \to x$
negative_passages.docid: 134450
negative_passages.text: Chinese Remainder Theorem $\ x\equiv m_k-2\pmod{m_k}$
negative_passages.docid: 63183
negative_passages.text: Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
negative_passages.docid: 145722
negative_passages.text: Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
negative_passages.docid: 60804
negative_passages.text: About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
negative_passages.docid: 77064
negative_passages.text: Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
negative_passages.docid: 122741
negative_passages.text: $x^2$ is congruent to $-1 \bmod p$
negative_passages.docid: 106584
negative_passages.text: If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
negative_passages.docid: 9806
negative_passages.text: $M$ finitely generated if submodule and quotient are finitely generated.
negative_passages.docid: 78852
negative_passages.text: If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
negative_passages.docid: 134365
negative_passages.text: When is this true: If $m^2$ is a multiple of N then m is a multiple of N