query: Mathematical induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
positive_passages.docid: 21100
positive_passages.text: Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$
negative_passages.docid: 962
negative_passages.text: divisibility problem using induction
negative_passages.docid: 22592
negative_passages.text: $1\cdot 3 + 2\cdot 4 + 3\cdot 5 + \cdots + n(n+2) = n(n+1)(2n+7)/6$ by mathematical induction
negative_passages.docid: 121874
negative_passages.text: Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
negative_passages.docid: 50670
negative_passages.text: Use Mathematical Induction to prove 6 divides $n(n+1)(2n+1)$
negative_passages.docid: 65315
negative_passages.text: Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step)
negative_passages.docid: 122386
negative_passages.text: How do I prove that $3$ divides $14^{2n}-1$ by induction?
negative_passages.docid: 14435
negative_passages.text: Show by induction: $(1+\frac{1}{n})^{n}<n$
negative_passages.docid: 60073
negative_passages.text: Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$
negative_passages.docid: 17269
negative_passages.text: Strong Induction Proof: Fibonacci number even if and only if 3 divides index
negative_passages.docid: 16812
negative_passages.text: Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
negative_passages.docid: 151692
negative_passages.text: When does $2^n-1$ divide $3^n-1$?
negative_passages.docid: 56760
negative_passages.text: Prove by induction that an expression is divisible by 11
negative_passages.docid: 117969
negative_passages.text: Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$.
negative_passages.docid: 117380
negative_passages.text: Divisibility Rule for 9
negative_passages.docid: 156356
negative_passages.text: Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction