query: Use induction to prove that $6$ divides $n^3 - n$.
positive_passages.docid: 62100
positive_passages.text: Show that $n^3-n$ is divisible by $6$ using induction
negative_passages.docid: 77579
negative_passages.text: Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$
negative_passages.docid: 117969
negative_passages.text: Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$.
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 26912
negative_passages.text: Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
negative_passages.docid: 106502
negative_passages.text: Prove by induction that for the Fibonacci numbers $F(n)$ with $n \ge 6$, $F(n) \ge 2^{n/2}$
negative_passages.docid: 21100
negative_passages.text: Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
negative_passages.docid: 122711
negative_passages.text: Prove by induction that $n^3 + 11n$ is divisible by $6$ for every positive integer $n$.
negative_passages.docid: 67527
negative_passages.text: Proof by Induction that $3^n ≥ 1+2^n$
negative_passages.docid: 124240
negative_passages.text: Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
negative_passages.docid: 120227
negative_passages.text: Proving $6^n - 1$ is always divisible by $5$ by induction
negative_passages.docid: 130488
negative_passages.text: Prove by induction that $3\mid n^3 - n$
negative_passages.docid: 75329
negative_passages.text: Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$
negative_passages.docid: 136700
negative_passages.text: Proving divisibility of expression using induction
negative_passages.docid: 10604
negative_passages.text: Prove $3^n > n^2$ by induction
negative_passages.docid: 100080
negative_passages.text: Proving with induction $(1-x)^n<\frac 1 {1+nx}$
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$