Question

How many integers n are there such that n+1 divides n2 +1

Answer 1

Answer:

n=1 because n must be a positive odd number and 2 is divisible by ( n+1).

Answer 2

Answer:

First note that n2 – 1 = (n + 1)(n – 1).

So, (n + 1)|n2 – 1 for all integer n except (–1).

Now, suppose (n+1)| (n2 + 1).

Therefore, n+1 | (n2 + 1) – (n2 – 1)=2

⇒ n+1 is a factor of 2.

So, possible values (n+1) can be 1, 2, (–1) or (–2).

Therefore, possible values of n are 0, 1, (–2) or (–3).

Verify that each of them satisfies the given condition.