query: Given positive integers a and b such that $a\mid b^2, \:b^2\mid a^3, \:a^3\mid b^4,\: b^4\mid a^5$..., prove $a=b.$
positive_passages.docid: 21006
positive_passages.text: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
negative_passages.docid: 44309
negative_passages.text: Prove $5a^2 ≠ 3b^2$ for all non-zero rational numbers $a$ and $b$
negative_passages.docid: 85977
negative_passages.text: Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
negative_passages.docid: 4141
negative_passages.text: Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
negative_passages.docid: 127891
negative_passages.text: If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
negative_passages.docid: 26062
negative_passages.text: Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
negative_passages.docid: 63931
negative_passages.text: When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.
negative_passages.docid: 5981
negative_passages.text: For which values of $a$ does $a-2 \mid a^3+4$
negative_passages.docid: 46286
negative_passages.text: Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.
negative_passages.docid: 115866
negative_passages.text: Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
negative_passages.docid: 114185
negative_passages.text: Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 142133
negative_passages.text: Prove that if $a^2 + b^2 = c^2$, then 5 divides abc. a,b,c are positive integers.
negative_passages.docid: 12954
negative_passages.text: Prove if $ac>bc$ and $c>0$, then $a>b$
negative_passages.docid: 111267
negative_passages.text: Prove that: $2^a+3^b<3a+4b$
negative_passages.docid: 127293
negative_passages.text: Prove that $(a-b) \mid (a^n-b^n)$