Prove: for any natural number n, either n is prime, n is a perfect square, or n|(n-1)!

Suppose that a natural number n is neither a prime nor a perfect square. Then, there exist natural numbers p and q such that p < n, q < n , p ≠ q and pq = n.

Now, note that (n - 1)! = (n - 1)(n - 2)(n - 3)...(3)(2)(1).

Since p < n and q < n, p = (n - k) and q = (n - l) where k ≠ l and k, l ∈ {∈1, 2, 3, ... , n - 1}. But we know that (n - k)(n - l) | (n - 1)!

Hence, pq | (n - 1)! and therefore n | (n - 1)!

Hence, any natural number n is either prime, a complete square or divides (n - 1)!, as required.

konichiwa~! my name is zalgo and i am here to answer your anime-related question today. more than likely, kirishima would kill izuku, marry mino and kiss momo. all of this would be my opinion by the way. (and i am also a weeb lol)

i hope this you! : d

stay and stay proud! " - zalgo

(and do you think you can mark me as brainliest? i'd greatly appreciate it! arigato~! )

stethere all not eveanp-by-step explanation: