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)?

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 CANT",. The long answer includes the explanation and (as an option) the proof.

Most of the solvers got it right.

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