query: How to prove or disprove that $\{ 12x + 25y | x , y \in \mathbb{Z} \} = \mathbb{Z}$.
positive_passages.docid: 25907
positive_passages.text: Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
negative_passages.docid: 108801
negative_passages.text: Given that $xyz=1$, prove that $\frac{x}{1+x^4}+\frac{y}{1+y^4}+\frac{z}{1+z^4}\le \frac{3}{2}$
negative_passages.docid: 106965
negative_passages.text: How to algebraically prove $\binom{n+m}{2} = nm + \binom{n}{2} + \binom{m}{2}$?
negative_passages.docid: 105263
negative_passages.text: Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
negative_passages.docid: 111967
negative_passages.text: Let $p$ be a prime number. Prove that there exists $x∈Z$ for which $p|x^2−x+ 3$ if and only if there exists $y∈Z$ for which $p|y^2−y+ 25$.
negative_passages.docid: 17677
negative_passages.text: Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$
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: 39492
negative_passages.text: Finding $z\in \mathbb C$ with $\sin z = 2$?
negative_passages.docid: 41383
negative_passages.text: Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
negative_passages.docid: 126429
negative_passages.text: Let $R=\mathbb{Z}\left[\sqrt{−5}\right]$. Let $I=\left(2,1+\sqrt{−5}\right)$. Show $I^2 = (2)$ in $R$.
negative_passages.docid: 130027
negative_passages.text: Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$
negative_passages.docid: 93029
negative_passages.text: Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
negative_passages.docid: 46106
negative_passages.text: Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$
negative_passages.docid: 76049
negative_passages.text: Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
negative_passages.docid: 110256
negative_passages.text: Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
negative_passages.docid: 152643
negative_passages.text: $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
negative_passages.docid: 5595
negative_passages.text: $x^y = y^x$ for integers $x$ and $y$