query: Prove that if $n$ is composite, then $(n-1)! \equiv 0 \pmod n$
positive_passages.docid: 9319
positive_passages.text: If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.
negative_passages.docid: 78356
negative_passages.text: How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$
negative_passages.docid: 141088
negative_passages.text: Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
negative_passages.docid: 59462
negative_passages.text: Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
negative_passages.docid: 47060
negative_passages.text: Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 148458
negative_passages.text: $x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
negative_passages.docid: 153763
negative_passages.text: Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
negative_passages.docid: 2218
negative_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 118140
negative_passages.text: Prove that $ \frac{n(n-1)(n-k+1)}{k(k-1)\cdots1} = \frac{n!}{(n-1)!k!}$
negative_passages.docid: 111951
negative_passages.text: Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
negative_passages.docid: 9522
negative_passages.text: Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
negative_passages.docid: 27147
negative_passages.text: Proving $n! > n^3$ for all $n > a$
negative_passages.docid: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 9787
negative_passages.text: Primitive roots modulo primes congruent to n!
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
negative_passages.docid: 94855
negative_passages.text: Prove if a (mod n) has a multiplicative inverse, then it's unique