query: If $3\mid a$, then $3\mid a^2$
positive_passages.docid: 153313
positive_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 50534
negative_passages.text: Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.
negative_passages.docid: 11857
negative_passages.text: $A$ be a complex $3\times 3$ matrix such that $A^3=-I$
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
negative_passages.docid: 87522
negative_passages.text: If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
negative_passages.docid: 30583
negative_passages.text: If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 13914
negative_passages.text: If $a^2+b^2=c^2$ then $a$ or $b$ is even.
negative_passages.docid: 118574
negative_passages.text: prove that $24 \mid a(a^2-1)$
negative_passages.docid: 77164
negative_passages.text: If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
negative_passages.docid: 15191
negative_passages.text: If $a^2$ divides $b^2$, then $a$ divides $b$
negative_passages.docid: 104498
negative_passages.text: Show that $x^3y^3(x^3+y^3) \leq 2$
negative_passages.docid: 89184
negative_passages.text: Prove $30|(a^3b-ab^3) $
negative_passages.docid: 11693
negative_passages.text: If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
negative_passages.docid: 62770
negative_passages.text: Does $a\mid bc$ imply $a\mid b$ or $a\mid c$?
negative_passages.docid: 69874
negative_passages.text: Are there any integer solutions to $a^3=b^2$?