query: Prove by induction that $n^3-n$ is divisible by $24$ for all odd positive integers
positive_passages.docid: 17965
positive_passages.text: Prove 24 divides $u^3-u$ for all odd natural numbers $u$
negative_passages.docid: 50670
negative_passages.text: Use Mathematical Induction to prove 6 divides $n(n+1)(2n+1)$
negative_passages.docid: 63416
negative_passages.text: Prove $(n^5-n)$ is divisible by 5 by induction.
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: 8366
negative_passages.text: Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
negative_passages.docid: 55273
negative_passages.text: Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
negative_passages.docid: 153123
negative_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 85941
negative_passages.text: Proof by induction that $n^3 - n$ is divisible by $6$
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: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
negative_passages.docid: 17269
negative_passages.text: Strong Induction Proof: Fibonacci number even if and only if 3 divides index
negative_passages.docid: 130488
negative_passages.text: Prove by induction that $3\mid n^3 - n$
negative_passages.docid: 11216
negative_passages.text: Induction: How to prove that $ab^n+cn+d$ is divisible by $m$.
negative_passages.docid: 153281
negative_passages.text: Prove by induction that $3^{2n+3}+40n-27$ is divisible by 64 for all n in natural numbers
negative_passages.docid: 106847
negative_passages.text: Proof that $n^3+2n$ is divisible by $3$