I WONDER (Posted on 2011-01-13)

	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.

	Subject	Author	Date
	re: Solution	Ady TZIDON	2011-01-28 01:55:45
	Solution	hoodat	2011-01-14 21:01:30
	re(2): Playing.	brianjn	2011-01-14 07:40:01
	re: Playing.	Ady TZIDON	2011-01-14 04:15:45
	Playing.	brianjn	2011-01-14 03:52:14
	re: Hmmmmm...	Ady TZIDON	2011-01-14 02:05:59
	Hmmmmm...	Math Man	2011-01-13 21:18:53
	Divisibility in any base.	Jer	2011-01-13 15:28:43
	re: my solution	Charlie	2011-01-13 14:24:15
	my solution	Daniel	2011-01-13 12:14:32