query: Prove or disprove $a^{10}=b^{10} \pmod{10\alpha}$
positive_passages.docid: 25299
positive_passages.text: What is the remainder when $2^{1990}$ is divided by $1990$?
negative_passages.docid: 157771
negative_passages.text: Nested exponent modulus, $2^{2^{517}} \pmod {23}$
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
negative_passages.docid: 147982
negative_passages.text: Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves "
negative_passages.docid: 38727
negative_passages.text: Modular Exponentiation and Primes
negative_passages.docid: 17176
negative_passages.text: Proof that $x^n \mod b = (x \mod b)^n$
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 43567
negative_passages.text: Some hints for "If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$"
negative_passages.docid: 24785
negative_passages.text: How prove this equation $\alpha=\aleph_{0}\alpha$
negative_passages.docid: 101159
negative_passages.text: Prove If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
negative_passages.docid: 115176
negative_passages.text: Can we say that $\left(a+b\right)^{\alpha}>a^{\alpha}+b^{\alpha}$ for all $a,b>0$ and $\alpha>1$?
negative_passages.docid: 135986
negative_passages.text: Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 102990
negative_passages.text: Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$
negative_passages.docid: 104484
negative_passages.text: Prove that $r$ is a primitive root modulo $p$ if and only if $r^{(p−1)/q}\not\equiv 1\pmod{p}$
negative_passages.docid: 66238
negative_passages.text: Proving $\mathbb{Z}[\sqrt {10}]$ is not a UFD
negative_passages.docid: 99631
negative_passages.text: Prove/Disprove: Find $a\in S_4$ that is not a power of $b$ such that $ab=ba$ while $b=(123)$