query: Find all positive integers n such that $n + 3$ divides $n^2 + 27$
positive_passages.docid: 151017
positive_passages.text: Division in $1$ variable
negative_passages.docid: 88662
negative_passages.text: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
negative_passages.docid: 130488
negative_passages.text: Prove by induction that $3\mid n^3 - n$
negative_passages.docid: 9302
negative_passages.text: Questions about Divisibility of $2^n$ by $3$
negative_passages.docid: 66578
negative_passages.text: Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
negative_passages.docid: 157186
negative_passages.text: Finding integers satisfying $m^2 - n^2 = 1111$
negative_passages.docid: 56574
negative_passages.text: For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
negative_passages.docid: 78416
negative_passages.text: Show that $7$ divides $a^{15}-a^3$ for any integer $a$
negative_passages.docid: 72389
negative_passages.text: Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
negative_passages.docid: 27063
negative_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 114404
negative_passages.text: For which positive integer values of $n$ does $a|mn-b?$
negative_passages.docid: 106847
negative_passages.text: Proof that $n^3+2n$ is divisible by $3$
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: 27147
negative_passages.text: Proving $n! > n^3$ for all $n > a$
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$