"Among any $n+1$ integers not exceeding $2n$, one of the integers divides the other" isn't true? $n=10$ for example?

Question

There have been other SE posts on this question, but I haven't looked at them yet because I want to try to solve the problem myself first.

However, the problem makes no sense to me. For example, $n=10$. Then the problem is saying that among the numbers $11, 12,13,14,15,16,17,18,19,20$ one divides the other. This is clearly not true...

This must be a problem with my misunderstanding, since other people have seemingly understood and answered the problem successfully.

Note that the full phrasing of the question (or one version of it) is: "Show that among any $n+ 1$ positive integers not exceeding $2n$ there must be an integer that divides one of the other integers."

Side question: in this post: Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other, everyone seems to express "the integers between $n-1$ and $2n$" as "the $(n-1)$-subset of $[2n]$". How does the latter idea imply the former? Does $[2n]$ denote an equivalence class? Even then, this makes no sense to me.

Any help is appreciated.

You've only listed $10$ integers less than or equal to $20$ and the statement calls for at least $11$. — saulspatz, Oct 02 '19 at 00:44
@saulspatz Wait, I thought the statement was referring to all the integers between $n+1$ and $2n$, inclusive? Is it just referring to a set of $11$ integers that are all less than or equal to $2n$? — James Ronald, Oct 02 '19 at 00:45
It wouldn't be just 11 integers though - only for when n=10. It would just be generalized as n+1 integers. — Jaemin Kim, Oct 02 '19 at 01:00
And if you want a nice proof on this using mathematical induction, check out link — Jaemin Kim, Oct 02 '19 at 01:18

score 2 · Answer 1 · answered Oct 02 '19 at 02:49

2

IIRC, the answer goes something like this.

Write the $j$-th integer as $a_j = 2^{b_j}c_j $ where $c_j$ is odd.

Since there are $n$ odd integers in $1:2n$, and there are $n+1\ a_j$, two of the $c_j$ must be equal, say $c_u = c_v$.

Then $a_u | a_v$ if $b_u < b_v$ and $a_v | a_u$ if $b_v < b_u$.

This is not original.

answered Oct 02 '19 at 02:49

marty cohen

107,799

Thank you. Can you please explain how you wrote the jth integer as multiplication of even and odd? What about 5? 5 does not equal to multiplication of even and odd for example. – Avv Jun 28 '21 at 16:12
1

If the integer is odd, the exponent of 2 is 0. – marty cohen Jun 29 '21 at 00:47

"Among any $n+1$ integers not exceeding $2n$, one of the integers divides the other" isn't true? $n=10$ for example?

1 Answers1