Foundations, part 4

Number Theory Level 3

$\large (n - 1)!\equiv -1 \pmod n$ Given a natural number $n > 1$ such that the above relation is true.

We can conclude that $\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_} .$

n

pq

,where

p

and

q

are distinct primes.

n

p

,where

p

is a prime.

n

pqr

,where

p

q

and

r

are distinct primes.

n

p^k

, where

p

is prime, k>1.

1 solution

Md Zuhair
Apr 14, 2017

Relevant wiki: Wilson's Theorem

By Wilson's Theorem , for any integer $n= p$ [ p is prime]

We have

$(n-1)! = -1 \mod n$

Verify For $n=3$ , we have $2! \space \mod 3 \implies \space 2 \mod 3 = -1 \mod 3$