MathLabs

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 1 of 5: Name the quantity
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Fn=⌊(n−1)!n(n+1)⌋F_n=\left\lfloor\dfrac{(n-1)!}{n(n+1)}\right\rfloor
Detailed analysis

Let F_n denote the required floor. For n=1,2,3,4,5, the numerator is smaller than n(n+1), so F_n=0, which is even.