Library / Annual Harvard-MIT November Tournament /7 of 20

Annual Harvard-MIT November Tournament

Number theory Difficulty 4.6 AIME Prove it United States

Problem:
Find the sum of all positive integers $n$ such that $n$ divides $n^{2}+n+2$ .

Solution

Solution:
Since $n$ always divides $n^{2}+n$ , the only $n$ that work are divisors of $2$ , because if $n$ divides $a$ and $n$ divides $b$ , then $n$ divides $a+b$ . So the solutions are $1$ and $2$ which sum to $3$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.