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Interesting induction questions

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Mahan:
Hi all,
I just wanted to share an interesting induction question.The idea behind the question is not what we usually use to solve the majority of induction questions. So I thought, it would be interesting to try to solve this question.
 

RuiAce:
What exactly does every 2n+1 natural number mean here?

Mahan:

--- Quote from: RuiAce on October 30, 2016, 11:45:54 pm ---What exactly does every 2n+1 natural number mean here?

--- End quote ---

It is an induction question. It means if you pick any natural number n, i.e n=10, and if you choose 2^(n+1) natural numbers, you can always choose 2^(n) natural numbers from those 2^(n+1) such that the sum is divisible by 2^(n). I hope that clarifies the question. :)

RuiAce:

--- Quote from: Mahan on October 30, 2016, 11:49:33 pm ---It is an induction question. It means if you pick any natural number n, i.e n=10, and if you choose 2^(n+1) natural numbers, you can always can choose 2^(n) natural numbers from those 2^(n+1) such that the sum is divisible by 2^(n). I hope that clarifies the question. :)

--- End quote ---
And these 2n+1 numbers have no upper bound? So we're allowed to pick any 2n+1 elements of the natural numbers we like?

So take n=10, then if we pick any 211 elements of the natural numbers (i.e. one of them could be say 212) then the sum of 210 of such elements must be divisble by 210?

Seems like a nice question but we need to clear up the technicalities

Mahan:

--- Quote from: RuiAce on October 30, 2016, 11:52:15 pm ---And these 2n+1 numbers have no upper bound? So we're allowed to pick any 2n+1 elements of the natural numbers we like?

So take n=10, then if we pick any 211 elements of the natural numbers (i.e. one of them could be say 212) then the sum of 210 of such elements must be divisble by 210?

Seems like a nice question but we need to clear up the technicalities

--- End quote ---
Yes that's correct.
2^10 is the number of natural numbers you can pick and there is no upper bound for the  natural numbers you chose.

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