query: Let $G=\{n \in \Bbb N \mid n\leq 55 , \; \gcd(n,55)=1\}$ be the group under multiplication modulo $55$. Find $x$ is $x^2=26$ and $x>30$
positive_passages.docid: 21530
positive_passages.text: How many solutions of $\mod 63$ : $x^2=1 \pmod7 $ and $x^3=1\pmod 9$
negative_passages.docid: 20823
negative_passages.text: If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
negative_passages.docid: 150958
negative_passages.text: Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
negative_passages.docid: 120642
negative_passages.text: Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
negative_passages.docid: 33996
negative_passages.text: Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
negative_passages.docid: 15168
negative_passages.text: If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
negative_passages.docid: 23066
negative_passages.text: If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
negative_passages.docid: 14693
negative_passages.text: $d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
negative_passages.docid: 1713
negative_passages.text: number theory $\gcd(a,bc)=\gcd(a,c)$
negative_passages.docid: 117468
negative_passages.text: $o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
negative_passages.docid: 147517
negative_passages.text: Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
negative_passages.docid: 100605
negative_passages.text: let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
negative_passages.docid: 57421
negative_passages.text: If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian?
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 7091
negative_passages.text: If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
negative_passages.docid: 151025
negative_passages.text: If G is a group of order n=35, then it is cyclic
negative_passages.docid: 69279
negative_passages.text: If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$