query: $1^n-3^n-6^n+8^n$ is divisible by $10$
positive_passages.docid: 17243
positive_passages.text: Remainder when $2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by $1998$
negative_passages.docid: 128306
negative_passages.text: Show that for all $n ∈ Z$, $2n^3 + 3n^2 + n$ is divisible by both 2 and 3.
negative_passages.docid: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
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
negative_passages.docid: 117167
negative_passages.text: $3^{n+1}$ divides $2^{3^n}+1$
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: 52032
negative_passages.text: Prove that ${n^5 - n}$ is divisible by 5
negative_passages.docid: 71625
negative_passages.text: Divisibility question
negative_passages.docid: 104965
negative_passages.text: Find $n$ such that polynomial is divisible
negative_passages.docid: 28957
negative_passages.text: Verifying that $2^{44}-1$ is divisible by $89$
negative_passages.docid: 112147
negative_passages.text: Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
negative_passages.docid: 10744
negative_passages.text: Prove that a number is divisible by 3 iff the sum of its digits is divisible by 3
negative_passages.docid: 11470
negative_passages.text: Divisibility of integers
negative_passages.docid: 106847
negative_passages.text: Proof that $n^3+2n$ is divisible by $3$
negative_passages.docid: 15631
negative_passages.text: Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$.
negative_passages.docid: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
negative_passages.docid: 29963
negative_passages.text: $6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$