$m$ is a positive integer, and $ m > 1$,
Prove that if $(m - 1)! + 1$ is divisible by $m$, $m$ is a prime.
Solve this by making a contradiction.
My english isn't so well. Please help and thank you for your attention :)
$m$ is a positive integer, and $ m > 1$,
Prove that if $(m - 1)! + 1$ is divisible by $m$, $m$ is a prime.
Solve this by making a contradiction.
My english isn't so well. Please help and thank you for your attention :)
Let $m$ be a positive integer $> 1$. Show that if $(m-1)! + 1$ is divisible by $m$, then $m$ is a prime.
HINTS
If $m$ is not a prime, then it obviously contains some prime factor $m'$ such that $1
If m is not a prime, it has a prime divisor $p
But as $p\mid m$ and $m\mid((m-1)!+1)$ so, $p\mid((m-1)!+1)$
$\implies p\mid((m-1)!+1-(m-1)!)$ as $q\mid a$ and $q\mid b\implies q\mid (ax+by)$ where $a,b$ are any integers.
$\implies p\mid 1\implies m$ can not have divisor p such that $1 , hence $m$ must be prime.
If $m$ divides $(m-1)!+1$, there exists $k \in \mathbb{Z}$ such that $(m-1)!+1=km$ so $-(m-1)!+km=1$. From Bézout's identity, you deduce that for all $n