Prove that there exists $n_0 \in \mathbb{Z}$ such that $I = \{kn_0 | k \in \mathbb{Z}\}$ given that $I$ satisfies the following properties
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
43010
Prove that for every n ∈ Z, there exist a, b ∈ Z such that $n = 5a + 2b$.
107010
How many elements of order $k$ are in $S_n$?
102112
Idempotents in $\mathbb Z_n$
152104
How do I prove that $ij = k?$
46039
Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
119648
For which $n$ and $k$ are there nontrivial homomorphisms from $S_n$ to $\mathbb{Z}/k\mathbb{Z}$?
3786
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
157924
$\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
127843
Prove that, for any natural number $k$ $\exists m \in M $ s.t i) $m$ has exactly $k$ $1$'s, an
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
116943
Proof for maximal ideals in $\mathbb{Z}[x]$
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
157597
Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
96903
How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
3599
show that if ${a_n} \to a$, and $a>0$, then $\exists N$ such that $a_n>0$ for $n \ge N$.
92740
prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$