query: Show that if a positive integer $ n $ is composite then $ R(n) = \frac{10^{n}-1}{9}= 111...11 (n times) $ is composite
positive_passages.docid: 23932
positive_passages.text: Why do repunit primes have only a prime number of consecutive $1$s?
negative_passages.docid: 96027
negative_passages.text: Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
negative_passages.docid: 149508
negative_passages.text: Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
negative_passages.docid: 96062
negative_passages.text: Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
negative_passages.docid: 14815
negative_passages.text: How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
negative_passages.docid: 59638
negative_passages.text: How to prove this result on binomial coefficients?
negative_passages.docid: 68824
negative_passages.text: Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
negative_passages.docid: 1507
negative_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial
negative_passages.docid: 26588
negative_passages.text: Prove by induction that $10^n -1$ is divisible by 11 for every even natural number
negative_passages.docid: 50927
negative_passages.text: If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
negative_passages.docid: 115145
negative_passages.text: Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits
negative_passages.docid: 112907
negative_passages.text: Math Induction Proof: $(1+\frac1n)^n < n$
negative_passages.docid: 72672
negative_passages.text: Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
negative_passages.docid: 95141
negative_passages.text: $n^5-n$ is divisible by $10$?
negative_passages.docid: 4610
negative_passages.text: Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
negative_passages.docid: 34954
negative_passages.text: If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.