query: Proof: $m,n$ are integers: either $4$ divides $mn$ or else $4$ does not divide $n$.
positive_passages.docid: 143248
positive_passages.text: What approach should I take to establish this logical proof?
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: 88662
negative_passages.text: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
negative_passages.docid: 111951
negative_passages.text: Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
negative_passages.docid: 81778
negative_passages.text: If $m|n$ and $a$ is a primitive root of $n$, show that $a$ is a primitive root of $m$ (understanding a tip)
negative_passages.docid: 81267
negative_passages.text: A simple doubt in number theory problem.
negative_passages.docid: 148455
negative_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: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
negative_passages.docid: 72591
negative_passages.text: Find a bijection to prove a fact about integer partitions
negative_passages.docid: 71650
negative_passages.text: Proving by strong induction for a sequence of integers, $2^n$ divides term $n$
negative_passages.docid: 24098
negative_passages.text: Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 31555
negative_passages.text: Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 112191
negative_passages.text: How can I show $q^k$ does not divide ${{n}\choose{q}}$ when $q$ is a prime factor of $n$ and $k$ is the largest integer such that $q^k$ divides $n$?
negative_passages.docid: 96169
negative_passages.text: $n$ divides a product of $n$ consecutive integers (by $n$ divides one of them)
negative_passages.docid: 27544
negative_passages.text: With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$