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

Home > Numbers
Divisible by 7 (Posted on 2007-02-15) Difficulty: 3 of 5
Prove that if A = ½(15+√197), then 7 is a factor of [A^n] for all positive integer values of n, where [w] denotes the greatest integer less than or equal to w.

See The Solution Submitted by Dennis    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Comment 5 of 5 |
Let A=(15+sqrt197)/2 and B=(15-sqrt197)/2.
(A+B)^n=sum_i=0...n{(n over i)*A^i*B^(n-i)} = A^n + B^n + C.
From the first post we know that C = 7*k for some k, because it's a sum of numbers of the form A*B*d (for some integer d), and A*B is divisible by 7.
On the other hand we know that (A+B)^n=15^n, and 15%7=1, hence (15^n)%7=1 for all n>=0.
This shows that (A^n+B^n)%7=1. The last step is too notice, that 0<B<1, so 0 Nice one.
  Posted by Arek on 2007-03-05 07:26:24
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 (4)
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