query: The number of elements in $\mathbb{Z}_{11}$ satisfies $x^{12}-x^{10}=2$.
positive_passages.docid: 18752
positive_passages.text: How many elements $x$ in the field $\mathbb{ Z}_{11}$ satisfy the equation $x^{12} - x^{10} = 2$?
negative_passages.docid: 33645
negative_passages.text: Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$.
negative_passages.docid: 2900
negative_passages.text: How many $3$-subsets of $\{1,2,\ldots,10\}$ contain at least one even and one odd integer?
negative_passages.docid: 97666
negative_passages.text: Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$
negative_passages.docid: 15631
negative_passages.text: Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$.
negative_passages.docid: 32278
negative_passages.text: Adjoining 1/2 to $\Bbb{Z}/12\Bbb{Z}$
negative_passages.docid: 15000
negative_passages.text: Show that number of elements in $(Z/nZ)^*$ is $\phi (n)$
negative_passages.docid: 4776
negative_passages.text: Prove that the power set of an $n$-element set contains $2^n$ elements
negative_passages.docid: 56760
negative_passages.text: Prove by induction that an expression is divisible by 11
negative_passages.docid: 115016
negative_passages.text: Prove that the number of $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|=26 $ equals 12.
negative_passages.docid: 51153
negative_passages.text: Number of solutions to $x^2\equiv a\pmod m$
negative_passages.docid: 92166
negative_passages.text: Number of odd binomial coefficients is a power of $2$
negative_passages.docid: 86807
negative_passages.text: The units of $\mathbb Z[\sqrt{2}]$
negative_passages.docid: 114262
negative_passages.text: Find the cardinality of the following set $S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0 \}$
negative_passages.docid: 22084
negative_passages.text: $\mathbb{Z}[i] / (2+3i)$ has 13 elements
negative_passages.docid: 69833
negative_passages.text: Number of elements not equal to their inverses is even number
negative_passages.docid: 1409
negative_passages.text: Proving that the given set has at most $2^{n-1}$ elements