Proving that there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$
49875
Question on Comaximal Ideals
50927
If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
13026
Prove that if AB=I_m, then rk(A)=rk(B).
59442
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
84021
How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
123891
There are finitely many maps on nonnegative integers satisfying $\phi(ab)=\phi(a)+\phi(b)$
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
111032
Proof of Sets involving the Cartesian Product
145397
How to prove that for every $i$, $a_i = i$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
29640
If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
102234
Show $a^p \equiv a \mod 2p$
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
18634
Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$