Thread: sets and relatively prime integers

1. sets and relatively prime integers

If $S$ is any set of $n + 1$ integers selected from $1, 2, 3,\cdot\cdot\cdot,2n + 1$,

prove that $S$ contains two relatively prime integers.
Prove that the result does not hold if $S$ contains only $n$ integers.

I think that the pigeonhole principle might be a way to prove this, but I don't know to determine what are the pigeons and what are the pigeonholes.

Thanks for the help!

2. Originally Posted by keityo
If $S$ is any set of $n + 1$ integers selected from $1, 2, 3,\cdot\cdot\cdot,2n + 1$,
prove that $S$ contains two relatively prime integers.
Prove that the result does not hold if $S$ contains only $n$ integers.
You need to notice that if $\{a,b\}\subseteq \{1,2,\cdots,2n+1\}$ then $\gcd(a,b)\le n-1$.

3. Why is then true?

I just had an idea: if $S$ has only n integers, then the set ${2,4,6,\cdot\cdot\cdot,2n}$ will have n integers, and all will be not relatively prime, which proves the second statement.

If $S$ has n+1 integers, then one of the integers must be odd because there can only be n even numbers in the set ${1,2,...,2n + 1}$ Thus, 2 and that odd number will be relatively prime, which proves the second statement.

Is this proof ok?
Thanks again.

4. Use the pigeon-hole principle! Amongst the following $n$ sets, there must be one containing two of your numbers :

$\{1,2\}, \{3,4\},\hdots \{2n-1, 2n, 2n+1\}$

Now just remark that any two integers which lie in the same part of this partition are relatively prime.

5. For your second part: $\{2,4,6,8,...,2n\}$.