query: Let $a$ and $b$ be integers such that $a|b$ and $b|a$. Show that $a ∈ \{b,−b\}$.
positive_passages.docid: 24098
positive_passages.text: Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
negative_passages.docid: 153670
negative_passages.text: Prove that if $a \mid b$ and $b \mid c$, then $a \mid c$.
negative_passages.docid: 21006
negative_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: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 20878
negative_passages.text: If $(a,b)=1$ then prove $(a+b, ab)=1$.
negative_passages.docid: 125164
negative_passages.text: How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
negative_passages.docid: 76826
negative_passages.text: Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
negative_passages.docid: 33542
negative_passages.text: When is $A- B = B- A$?
negative_passages.docid: 23869
negative_passages.text: Prove the implication "if $ac>bc$ and $c>0$, then $a>b$" using the contrapositive
negative_passages.docid: 121700
negative_passages.text: If $a$ and $b$ are positive real numbers, then $a + b \geq 2 \sqrt{ab}$.
negative_passages.docid: 21726
negative_passages.text: Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
negative_passages.docid: 67483
negative_passages.text: Show that $|a-b|<\epsilon \Rightarrow |a|<|b|+\epsilon$
negative_passages.docid: 15168
negative_passages.text: If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
negative_passages.docid: 24160
negative_passages.text: Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$
negative_passages.docid: 96217
negative_passages.text: If a and b are non-negative real numbers then demonstrate inequality
negative_passages.docid: 59813
negative_passages.text: Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A