query: If $(a,n)=d>1$ and $k$ is any positive integer, then $a^{k} \not\equiv 1 \pmod n$
positive_passages.docid: 148455
positive_passages.text: Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
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: 31766
negative_passages.text: Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
negative_passages.docid: 44033
negative_passages.text: Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
negative_passages.docid: 66851
negative_passages.text: If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
negative_passages.docid: 38882
negative_passages.text: If $n$ is a positive integer such that $n^3 − n − 6 = 0$, then, for every positive integer $m$ with $m \not = n$, $m^3 − m − 6 \not = 0$.
negative_passages.docid: 32463
negative_passages.text: $ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
negative_passages.docid: 44897
negative_passages.text: Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$
negative_passages.docid: 149071
negative_passages.text: Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?
negative_passages.docid: 46741
negative_passages.text: Prove that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer without using concepts of abstract algebra
negative_passages.docid: 144017
negative_passages.text: How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$
negative_passages.docid: 153763
negative_passages.text: Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
negative_passages.docid: 40663
negative_passages.text: $a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$.
negative_passages.docid: 23879
negative_passages.text: If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$
negative_passages.docid: 84733
negative_passages.text: Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
negative_passages.docid: 5708
negative_passages.text: Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
negative_passages.docid: 141369
negative_passages.text: If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.