Show that there exists $i$ and $j$ with $1 \leq i<j \leq 51$ satisfying $a_i$ divides $a_j$.
2510
Prove that if 51 positive integers between 1 and 100 are chosen, then one of them must divide another.
77128
Necessary and Sufficient conditions for $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
106756
Union and Intersection of Sets $A_{i}$ and $B_{j}$
155886
let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$
54430
For $a,b$ coprime, there exists positive integers $x,y$ such that $ax-by=1$
141529
Proving existence of a minimum
125164
How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
22154
Show that given a set of positive n integers, there exists a non-empty subset whose sum is divisible by n
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
43246
Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
7802
Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer
125499
Find this sequence $a_{i}$ such $\gcd{(a_{i},a_{j})}=\gcd{(i,j)}$
6896
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
65534
Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
15191
If $a^2$ divides $b^2$, then $a$ divides $b$