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 2 of 5: Handle two consecutive composite numbers
Detailed analysis
Assume n≥8 and both n and n+1 are composite. Write each as a product of proper factors. All required factors are smaller than n and occur in (n-1)!; the exceptional small overlap n=8 is checked directly. Thus n(n+1) divides (n-1)!. Counting powers of 2 in the factorial leaves at least one factor 2 after division, so F_n is an even integer.