Find all integers n such that n^2 + 1 is divisible by n + 1 скачать в хорошем качестве

Find all integers n such that n^2 + 1 is divisible by n + 1

This proof explains how to find all integers n such that n² + 1 is divisible by n + 1. The key idea is to rewrite the expression in a clever way so that the divisibility condition becomes simple to understand. We start by observing that n² + 1 can be written as (n² − 1) + 2. Using the identity n² − 1 = (n − 1)(n + 1), we rewrite the expression as (n − 1)(n + 1) + 2. Now, when we divide the entire expression by n + 1, the first part divides perfectly, leaving: n − 1 + 2 / (n + 1) For the whole expression to be an integer, the remaining fraction 2 / (n + 1) must also be an integer. This means that n + 1 must divide 2. The only integers that divide 2 are ±1 and ±2. So we solve: n + 1 = 1 n + 1 = −1 n + 1 = 2 n + 1 = −2 This gives the solutions: n = 0 n = −2 n = 1 n = −3 Therefore, the only integers that satisfy the condition are: n = 0, 1, −2, −3 By rewriting the expression strategically and reducing the problem to a simple divisibility condition, we are able to find all possible integer solutions in a clear and systematic way.