Prove that $M/N$ and $N$ are finitely generated $\Rightarrow M$ is finitely generated.
9806
$M$ finitely generated if submodule and quotient are finitely generated.
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
59459
Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality
111951
Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
37012
The number of summands $\phi(n)$
16643
A slick proof that a field which is finitely generated as a ring is finite
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
135018
Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
47709
If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
95240
Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.
59134
Proving idempotent elements in $\mathbb{Z}_n$
18018
Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
32795
Proving $ \bigcup_{i=1}^n A_{i} \text{ is finite.} $ by Induction.
9846
Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]