query: Suppose $n$ is a positive integer. Decide the remainder of $x^{2n}-x^{n}$ when divided by $2$
positive_passages.docid: 26011
positive_passages.text: To show that $2$ | $x^{2n} - x^n$, $n \in \mathbb{Z^+}$ is the following adequate?
negative_passages.docid: 49744
negative_passages.text: Find remainder of division of $x^3$ by $x^2-x+1$
negative_passages.docid: 151692
negative_passages.text: When does $2^n-1$ divide $3^n-1$?
negative_passages.docid: 132150
negative_passages.text: How to show the full remainder in polynomial long division?
negative_passages.docid: 100336
negative_passages.text: Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
negative_passages.docid: 80502
negative_passages.text: $p(x)$ divided by $x-c$ has remainder $p(c)$? [Polynomial Remainder Theorem]
negative_passages.docid: 97805
negative_passages.text: $n+1$ is a divisor of $\binom{2n} {n}$
negative_passages.docid: 49372
negative_passages.text: $x+1/x$ an integer implies $x^n+1/x^n$ an integer
negative_passages.docid: 95033
negative_passages.text: Find the remainder of a number with a large exponent
negative_passages.docid: 128749
negative_passages.text: Taylor's Remainder $x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}<\log(1+x) <x-\frac{x^2}{2}+\frac{x^3}{3}$
negative_passages.docid: 110460
negative_passages.text: find remainder of a number
negative_passages.docid: 142769
negative_passages.text: Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.
negative_passages.docid: 31555
negative_passages.text: Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
negative_passages.docid: 142992
negative_passages.text: Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
negative_passages.docid: 148906
negative_passages.text: Finding integers $m,n$ such that $(m-n)^2=4mn/(m+n-1)$
negative_passages.docid: 112631
negative_passages.text: Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$
negative_passages.docid: 115264
negative_passages.text: Find when three numbers have the same remainder when divided by the same number