Problem 4
For a real number x, let floor(x) be the greatest integer not exceeding x. Prove that floor((n-1)! divided by n(n+1)) is even for every positive integer n.
Step 3 of 5: Use Wilson when n is prime
Detailed analysis
Let n be an odd prime, with the small case n=3 already handled. Since n+1 is composite, n+1 divides (n-1)! for n≥7. Wilson gives (n-1)!+1 divisible by n, hence (n-1)!+n+1 divisible by n as well; it is also divisible by n+1, so K=((n-1)!+n+1)/(n(n+1)) is an integer. The factorial has strictly more powers of 2 than n+1, so K is odd. The quotient in the problem is K-1/n, whose floor is K-1, even.