query: Let $n$ be an odd positive integer. Show that $24 \vert(n^3-n).$
positive_passages.docid: 17965
positive_passages.text: Prove 24 divides $u^3-u$ for all odd natural numbers $u$
negative_passages.docid: 1895
negative_passages.text: How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
negative_passages.docid: 78190
negative_passages.text: How to prove that for all natural numbers, $4^n > n^3$?
negative_passages.docid: 157294
negative_passages.text: Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
negative_passages.docid: 77383
negative_passages.text: Prove that $n^3=n \text{ mod }6$ for every integer $n$.
negative_passages.docid: 77564
negative_passages.text: How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 4077
negative_passages.text: Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
negative_passages.docid: 8366
negative_passages.text: Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
negative_passages.docid: 44391
negative_passages.text: How can I prove $n - 2\sqrt{n} = \Theta(n)$
negative_passages.docid: 105564
negative_passages.text: If $n$ is a positive integer greater than 1 such that $3n+1$ is perfect square, then show that $n+1$ is the sum of three perfect squares.
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 124816
negative_passages.text: Using the fact that $n^3 + 5n$ is divisible by $6$, infer that $n^4 - 2n^3 + 11n^2 + 14n$ is always divisible by $24$
negative_passages.docid: 133017
negative_passages.text: Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$