Question

If $n>1$ and $n$ divides $(n-1) !+1$, then

• A. $n$ must be prime
• B. $n$ must be divisible by exactly two primes
• C. $n$ must be a composite number
• D. none of these.

Answer 1

Answer: (A)

If $n$ is not prime, then there exists $r \in N$ such that $2 \leq r \leq n-1$ and $r \mid n$.
As $r \mid n$ and $n ![(n-1) !+1]$, we get $r ![(n-1) !+1]$
As $2 \leq r \leq n-1, r \mid(n-1)$ !, therefore $r \mid 1$. A contradiction.