query: How could I show that if $a$ is an integer, then $a^3 \equiv 1, 0, 6 \mod 7$?
positive_passages.docid: 156577
positive_passages.text: The cube of any number not a multiple of $7$, will equal one more or one less than a multiple of $7$
negative_passages.docid: 68960
negative_passages.text: How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
negative_passages.docid: 103499
negative_passages.text: $x^2 \equiv a \mod p$, $\ x^2 \equiv b \mod p$, and $x^2 \equiv ab \mod p$, prove either all three are solvable or exactly one
negative_passages.docid: 89184
negative_passages.text: Prove $30|(a^3b-ab^3) $
negative_passages.docid: 42780
negative_passages.text: Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 90438
negative_passages.text: To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 48875
negative_passages.text: $n^3 \equiv n^5 \pmod{12} $?
negative_passages.docid: 110752
negative_passages.text: $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
negative_passages.docid: 27147
negative_passages.text: Proving $n! > n^3$ for all $n > a$
negative_passages.docid: 40968
negative_passages.text: show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
negative_passages.docid: 46147
negative_passages.text: Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
negative_passages.docid: 85287
negative_passages.text: Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
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: 152039
negative_passages.text: Prove that $a^{2^n}=1 \mod 2^{n+2}$