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

Home > Logic
I WONDER (Posted on 2011-01-13) Difficulty: 4 of 5

Read carefully, then post your answer.


Can you provide a simple YES or NO correct answer to the question :
Is 1010101......101 (n ones interwoven with n-1 zeroes) evenly divisible (i.e. without remainder) by 111…1(a string of n ones)?

Rem: n>1

  Submitted by Ady TZIDON    
Rating: 4.0000 (3 votes)
Solution: (Hide)
The problem can be divided into 2 parts:
1.Preamble P:
Can you provide a simple YES or NO correct answer
  2.A question Q
Is 1010101......101 (n ones interwoven with n-1 zeroes) evenly divisible (i.e. without remainder) by 111…1(a string of n ones)?

    Since 101 does not divide 11 and 10101 divides 111, the question Q cannot be answered by YES(not always true) or NO(correct in truthfulness, but contradictory by being a single word).
It can be shown **(but it is not necessary in the context of the solution) that it is YES for odd number of ones and NO for even.

**  Q - solution
  Let A=10101……01 & B=11111…1, both consist of n ones. To find whether A/B is an integer, just multiply both numerator and denominator by 11.
Clearly A/B=11*A/11*B= 111...111/11*B
The number 111…111 consists of 2n ones and therefore is concatenation of B following B, equaling B*(10^n+1).
So the original Q boils down to deciding whether (10^n+1) divides 11.
It does for odd n , it does not for even.

Rem: If worded: Answer yes/no the question Q - you get a paradox, since by saying NO you answer in a single word and by answering YES you err.
The way it is posted (P+Q) the possible correct short answers are: "NO", or "NO, YOU CAN’T",. The long answer includes the explanation and (as an option) the proof.

Most of the solvers got it right.
     

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Solutionre: SolutionAdy TZIDON2011-01-28 01:55:45
Solutionhoodat2011-01-14 21:01:30
re(2): Playing.brianjn2011-01-14 07:40:01
re: Playing.Ady TZIDON2011-01-14 04:15:45
SolutionPlaying.brianjn2011-01-14 03:52:14
Hints/Tipsre: Hmmmmm...Ady TZIDON2011-01-14 02:05:59
Hmmmmm...Math Man2011-01-13 21:18:53
SolutionDivisibility in any base.Jer2011-01-13 15:28:43
Some Thoughtsre: my solutionCharlie2011-01-13 14:24:15
my solutionDaniel2011-01-13 12:14:32
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 (7)
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