Prove that $q$ divides (${}_q \mathrm{C}_x$) where $q$ is an odd prime and $x \in \Bbb{Z}$ is such that $1\lt x \lt q$.
5841
Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
96027
Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
27784
Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
80144
Prove that $x-1$ divides $x^n-1$
42932
Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
144622
Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
85525
Prove that $x^3+x+1$ splits in $Z_2[x]/(x^3+x+1)$
31555
Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
140948
Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds.
40131
If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24.
21439
For which $k\in \Bbb N$ does $x^2+x+1$ divide $ (x + 1)^{2k}+x^{2k} + 1$?
109778
Proving $(x-c)|(p(x)-p(c))$
83407
How to show that that $q \mid p-1$?
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
81601
Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.
31724
How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$?