query: Prove that If x and y are integers and x + y is odd , then x−y is odd.
positive_passages.docid: 142724
positive_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: 103116
negative_passages.text: Prove that any odd number can be a leg in a Pythagorean Triple
negative_passages.docid: 77564
negative_passages.text: How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?
negative_passages.docid: 96097
negative_passages.text: Let $k$ be an odd positive integer. Prove that $(1+2+3...+n)|(1^k+2^k+3^k+...+n^k)$ for all positive integers $n$.
negative_passages.docid: 118727
negative_passages.text: $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.
negative_passages.docid: 135750
negative_passages.text: Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
negative_passages.docid: 63774
negative_passages.text: If a and b are odd integers, then $8\mid (a^2-b^2)$
negative_passages.docid: 86211
negative_passages.text: If $x+y+z \geq xyz$ Prove that $x^2 + y^2 + z^2 \geq xyz$.
negative_passages.docid: 25825
negative_passages.text: Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 121353
negative_passages.text: If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$
negative_passages.docid: 123267
negative_passages.text: Proving that even numbers equal the sum of two odd numbers.
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 74713
negative_passages.text: Show that for any natural numbers, if $S(x)=S(y)$ then $x=y$.
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 82742
negative_passages.text: Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
negative_passages.docid: 31895
negative_passages.text: Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$