All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Oblong Difference Observation (Posted on 2015-10-15) Difficulty: 3 of 5
Find all possible pairs (M, N) of oblong numbers that satisfy:

M - N = 2016

Prove that there are no others.

*** As an extra challenge solve this puzzle without using a computer program aided method.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Full analytical solution Comment 3 of 3 |
let N = n(n+1)
let M = m(m+1) where m = n+k

Then 2016 = M - N = k(2n + k + 1) after substituting and simplifying.

1) Clearly k < sqrt(2016), so k <= 44
2) k and (2n+k+1) are factors of 2016, which is 2^5 * 3^2 * 7
3) If k is even then (2n + k +1) is odd and therefore has no factors of 2.
   So, if k is even it must be a multiple of 32.  32 is the only multiple under 44
4) Every odd k under 44 gives rise to a solution, namely n = (2016/k - k - 1)/2.
   k can be 1, 3, 7, 9 or 21
   
So, there are only 6 solutions, corresponding to k = 1, 3, 7, 9, 21 and 32.
These are the same 6 that Charlie found

  Posted by Steve Herman on 2015-10-15 17:42:35
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information