Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2021 WSMO Accuracy Round Problems/Problem 1

Revision as of 11:34, 23 December 2021 by Pinkpig (talk | contribs) (Created page with "==Problem== Find the sum of all the positive integers <math>n</math> such that <math>n</math> is <math>\frac{2n^2-5n+5}{n-5}</math> an integer. ==Solution== Note that <math>2...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Find the sum of all the positive integers $n$ such that $n$ is $\frac{2n^2-5n+5}{n-5}$ an integer.

Solution

Note that $2n^2-5n+5=(n-5)(2n+5)+30.$ The given condition implies that $\frac{(n-5)(2n+5)+30}{n-5}=2n+5+\frac{30}{n-5},$ which implies that $n-5$ is a factor of 30. Thus, \[n-5=-30,-15,-10,-6,-5,-3,-2,-1,1,2,3,5,6,10,15,30.\] Solving, we find that \[n=-25,-10,-5,0,2,3,4,6,7,8,10,11,15,20,35.\] Since $n$ is positive, we find that $n=2,3,4,6,7,8,10,11,15,20,35.$ Summing, we find that the answer is \[2+3+4+6+7+8+10+11+15+20+35=\boxed{121}.\]

~pinkpig

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_WSMO_Accuracy_Round_Problems/Problem_1&oldid=168377"