If n1 and n divides (n-1) !+1, then

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.

Tardigrade Admin · Accepted Answer

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

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

$n$ must be prime

$n$ must be divisible by exactly two primes

$n$ must be a composite number

none of these.

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

Q. If n>1 and n divides (n−1)!+1, then

n must be prime

n must be divisible by exactly two primes

n must be a composite number