query: Prove that $n^3=n \text{ mod }6$ for every integer $n$.
positive_passages.docid: 26301
positive_passages.text: Prove that $x^3 \equiv x \bmod 6$ for all integers $x$
negative_passages.docid: 81058
negative_passages.text: Evaluate $6^{433} \pmod {21}$ and a proving question
negative_passages.docid: 38882
negative_passages.text: If $n$ is a positive integer such that $n^3 − n − 6 = 0$, then, for every positive integer $m$ with $m \not = n$, $m^3 − m − 6 \not = 0$.
negative_passages.docid: 82913
negative_passages.text: Is this a proof of Fermat's Last Theorem where n = 3?
negative_passages.docid: 32803
negative_passages.text: Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 56624
negative_passages.text: Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
negative_passages.docid: 92057
negative_passages.text: Prove that for every odd integer n, there is a group...
negative_passages.docid: 146851
negative_passages.text: Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$
negative_passages.docid: 136894
negative_passages.text: Reference request: Clean proof of Fermat's last theorem for $n=3$.
negative_passages.docid: 130488
negative_passages.text: Prove by induction that $3\mid n^3 - n$
negative_passages.docid: 94958
negative_passages.text: Prob. 6 (d), Chap. 1, in Baby Rudin, 3rd ed: How to complete this proof?
negative_passages.docid: 12365
negative_passages.text: Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
negative_passages.docid: 100892
negative_passages.text: Proving divisibility of $a^3 - a$ by $6$
negative_passages.docid: 124816
negative_passages.text: Using the fact that $n^3 + 5n$ is divisible by $6$, infer that $n^4 - 2n^3 + 11n^2 + 14n$ is always divisible by $24$
negative_passages.docid: 83935
negative_passages.text: Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$
negative_passages.docid: 127816
negative_passages.text: Number theory proof mod p