query: Show that if $a \equiv b\pmod{kn}$, then $a^k\equiv b^k\pmod{k^2n}$
positive_passages.docid: 17601
positive_passages.text: How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 22962
negative_passages.text: How to show $\binom{2p}{p} \equiv 2\pmod p$?
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 75166
negative_passages.text: Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
negative_passages.docid: 88036
negative_passages.text: Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
negative_passages.docid: 147253
negative_passages.text: Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
negative_passages.docid: 10236
negative_passages.text: Show that $a^n \mid b^n$ implies $a \mid b$
negative_passages.docid: 106767
negative_passages.text: $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
negative_passages.docid: 139503
negative_passages.text: How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
negative_passages.docid: 96329
negative_passages.text: Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$
negative_passages.docid: 131551
negative_passages.text: $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
negative_passages.docid: 87813
negative_passages.text: How to show that for all k, $k! \ge (k/2)^{k/2}$
negative_passages.docid: 101057
negative_passages.text: Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 33344
negative_passages.text: A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?