Problem: Find all positive integers n such that n + 1 is a factor of n² + 1.

Solution: Firstly, note that (n + 1)² = n² + 2n + 1 is always divisible by n + 1. So, if n² + 1 is divisible by n + 1, then (n + 1)² – (n² + 1) = 2n must also be divisible by n + 1. Furthermore, it is easy to see that 2(n + 1) = 2n + 2 is also divisible by n + 1. Since both 2n and 2n + 2 must be divisible by n + 1, 2n + 2 – 2n = 2 must also be divisible by n + 1. Hence n + 1 can only be equal to 1 or 2, as these are the only divisors of 2. Therefore n can only be equal to 1 or 0. Recall that n is a positive integer, so the only possible value of n is 1. Therefore, if n + 1 divides n² + 1, the only possible value of n is 1. Now all that is left is to check that 1 is actually a valid solution, as we cannot just come to that conclusion by only knowing that 1 is a possible solution. 1 + 1 = 2 is a factor of 1² + 1 = 2, hence 1 is indeed a valid solution.

Hence, we have shown that n = 1 is the only solution to this problem. This is a simple example of an argument in elementary number theory, and I hope you enjoyed working through it!!

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