query: Prove $n \in Z$ if $3|n$ and $8|n$ then $24|n$
positive_passages.docid: 20823
positive_passages.text: If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
negative_passages.docid: 925
negative_passages.text: Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
negative_passages.docid: 50433
negative_passages.text: Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$
negative_passages.docid: 90789
negative_passages.text: Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$
negative_passages.docid: 32166
negative_passages.text: Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
negative_passages.docid: 145891
negative_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 72511
negative_passages.text: If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
negative_passages.docid: 104855
negative_passages.text: Prove that $N \subset Z(G)$
negative_passages.docid: 1895
negative_passages.text: How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
negative_passages.docid: 1409
negative_passages.text: Proving that the given set has at most $2^{n-1}$ elements
negative_passages.docid: 55428
negative_passages.text: Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$
negative_passages.docid: 123155
negative_passages.text: If $mn+1$ is a multiple of $24$, prove that $m+n$ is also a multiple of 24.
negative_passages.docid: 41784
negative_passages.text: Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
negative_passages.docid: 106847
negative_passages.text: Proof that $n^3+2n$ is divisible by $3$
negative_passages.docid: 114780
negative_passages.text: $\forall a,b,c \in \mathbb{N}\!:\ a\mid b \Rightarrow ac\mid bc. $ Divisibility Proof
negative_passages.docid: 150156
negative_passages.text: Proving that $S_n$ has order $n!$
negative_passages.docid: 127400
negative_passages.text: Prove that ∀ n ∈ {n | n ∈ ℕ ∧ n > 7}: n = x*3 + y*5