Proof: Prove that for all n E N ... n and n+1 are co-prime. This is a proof so p
ID: 1941955 • Letter: P
Question
Proof: Prove that for all n E N ... n and n+1 are co-prime.This is a proof so please show all steps. Thank you
Explanation / Answer
So start with any n > 1 and write down one of its prime factors, say p. The prime factors of its successor, n + 1, are different from p. So there is at least some other prime, say q. Now consider the successor of the product n(n + 1). The prime factors of the latter are different from those of n and n + 1, p and q, in particular. Let r be one of those. Appply the same argument to the successor of n(n + 1)[n(n + 1) + 1] to obtain yet another prime, say s. Obviously the process can be extended indefinitely.
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