query: Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.
positive_passages.docid: 27371
positive_passages.text: A natural number multiplied by some integer results in a number with only ones and zeros
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 100305
negative_passages.text: Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff...
negative_passages.docid: 103784
negative_passages.text: Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)
negative_passages.docid: 140405
negative_passages.text: Show that there does not exist any $n$ for which $a^n=a$ forall $a\in \Bbb Z_m$ where $m$ is divisible by the square of some prime.
negative_passages.docid: 59479
negative_passages.text: For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 43010
negative_passages.text: Prove that for every n ∈ Z, there exist a, b ∈ Z such that $n = 5a + 2b$.
negative_passages.docid: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
negative_passages.docid: 24446
negative_passages.text: Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.
negative_passages.docid: 26514
negative_passages.text: Prove that if $n$ is a composite, then $2^n-1$ is composite.
negative_passages.docid: 103732
negative_passages.text: Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
negative_passages.docid: 108504
negative_passages.text: Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.
negative_passages.docid: 96268
negative_passages.text: Prove that $5$ divides $3^{3n+1}+2^{n+1}$
negative_passages.docid: 98963
negative_passages.text: Prove that neither $A$ nor $B$ is divisible by $5$
negative_passages.docid: 10191
negative_passages.text: Binomial theorem: divisibility by $n^2$
negative_passages.docid: 33866
negative_passages.text: Show that $n$ does not divide $2^n - 1$ where $n$ is an integer greater than $1$?