query: Let $x_1, x_2,\dots, x_n$ be an arrangement of the numbers $1, 2,\dots, n$. Show that if n is odd, then $(x_1 − 1)(x_2 − 2)\dots(x_n − n)$ is even.
positive_passages.docid: 151520
positive_passages.text: About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 52089
negative_passages.text: Group of $2n$ elements, $n$ odd, is not simple
negative_passages.docid: 100558
negative_passages.text: Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
negative_passages.docid: 18772
negative_passages.text: If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
negative_passages.docid: 6453
negative_passages.text: Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.
negative_passages.docid: 44484
negative_passages.text: Combinatorial identity $i{n \choose i } =n {n-1 \choose i - 1}$
negative_passages.docid: 123267
negative_passages.text: Proving that even numbers equal the sum of two odd numbers.
negative_passages.docid: 72054
negative_passages.text: Show that $\deg f$ is even when $n$ is odd.
negative_passages.docid: 134989
negative_passages.text: Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number
negative_passages.docid: 128834
negative_passages.text: Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
negative_passages.docid: 45037
negative_passages.text: Show $(2^m-1,2^n+1)=1$ if $m$ is odd
negative_passages.docid: 76405
negative_passages.text: Let n be an even integer. How many different ways can I choose subsets of the set {1,2...,n}
negative_passages.docid: 155784
negative_passages.text: If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
negative_passages.docid: 18953
negative_passages.text: Prove that even + odd is odd.
negative_passages.docid: 142724
negative_passages.text: Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
negative_passages.docid: 53194
negative_passages.text: Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even )