query: Suppose a set of integers $A$ such that $\lvert A \rvert = 501$ and for any $a \in A$, $1 \le a \le 1000$. Show some $a,b \in A,$ $a \vert b$
positive_passages.docid: 148803
positive_passages.text: 501 distinct coprime integers between 1 and 1000?
negative_passages.docid: 19046
negative_passages.text: Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
negative_passages.docid: 125011
negative_passages.text: Show lim sup (an) = lim inf (an) for increasing for increasing sequence of subsets.
negative_passages.docid: 132541
negative_passages.text: Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
negative_passages.docid: 67208
negative_passages.text: Can we say that there exist an integer $n$ such $A+nB$ invertible?
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 131983
negative_passages.text: Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
negative_passages.docid: 1409
negative_passages.text: Proving that the given set has at most $2^{n-1}$ elements
negative_passages.docid: 114185
negative_passages.text: Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$
negative_passages.docid: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 134770
negative_passages.text: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
negative_passages.docid: 28796
negative_passages.text: Use the Inclusion-Exclusion Principle to show that for any non-negative integers $m, r, n$ such that $m≤r≤n$...
negative_passages.docid: 22154
negative_passages.text: Show that given a set of positive n integers, there exists a non-empty subset whose sum is divisible by n
negative_passages.docid: 81267
negative_passages.text: A simple doubt in number theory problem.
negative_passages.docid: 144787
negative_passages.text: Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$
negative_passages.docid: 51706
negative_passages.text: If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
negative_passages.docid: 89989
negative_passages.text: A set of integers whose elements all divide $2015^{200}$ but do not divide each other