query: Prove that if $a$ is an integer, then $a(a^2+2)$ is divisible by $3$
positive_passages.docid: 151153
positive_passages.text: Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 62556
negative_passages.text: Prove $3|n(n+1)(n+2)$ by induction
negative_passages.docid: 38904
negative_passages.text: Prove that $(a + b + c)^{13}$ is divisible by $abc$ if $b|a^3$, $c|b^3$ and $a|c^3$.
negative_passages.docid: 41563
negative_passages.text: Prove that if the sum of digits of a number is divisible by 3, so is the number itself.
negative_passages.docid: 35666
negative_passages.text: How to prove the divisibility rule for $3\, $ [casting out threes]
negative_passages.docid: 94516
negative_passages.text: If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
negative_passages.docid: 85959
negative_passages.text: How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3$?
negative_passages.docid: 10604
negative_passages.text: Prove $3^n > n^2$ by induction
negative_passages.docid: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 34330
negative_passages.text: How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 90283
negative_passages.text: Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$
negative_passages.docid: 103496
negative_passages.text: Show that $3p^2=q^2$ implies $3|p$ and $3|q$
negative_passages.docid: 2522
negative_passages.text: Why is $a^n - b^n$ divisible by $a-b$?
negative_passages.docid: 11216
negative_passages.text: Induction: How to prove that $ab^n+cn+d$ is divisible by $m$.