anonymous
  • anonymous
Fool's problem of the day ( on request of asnaseer), probably easy , I haven't thought much, Let \(A = \{a_1, a_2 \cdots a_k\} \) be any set of \( k \) composite numbers such that \(1 \le a_i \le 120 \) for all \(i\) such that \(1 \le i \le k\). Find the least value of \(k\) such that there exists at least one pair \( (a_i, a_j) \), \(1\le  i, j \le  k\) in \(A\) which is not co prime ?
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
I should go to bed now, Enjoy guys :)
asnaseer
  • asnaseer
good night FFM
asnaseer
  • asnaseer
and thanks for the post

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anonymous
  • anonymous
You are always welcome :)
Mr.Math
  • Mr.Math
I guess k=4.
asnaseer
  • asnaseer
isn't it k=2 --> {1,2} are 1 and 2 considered to be co prime?
Mr.Math
  • Mr.Math
2 is not composite.
asnaseer
  • asnaseer
ah! - of course - thx for pointing that out Mr.Math
Mr.Math
  • Mr.Math
Neither is 1.
asnaseer
  • asnaseer
I thought 1 is coprime?
asnaseer
  • asnaseer
but not composite
Mr.Math
  • Mr.Math
Two integers a and b are said to be coprime (also spelled co-prime) or relatively prime if the only positive integer that evenly divides both of them is 1.
Mr.Math
  • Mr.Math
What about \(A=\{4,6\}?\), that's k=2.
asnaseer
  • asnaseer
but 2 divides 4 and 6?
Mr.Math
  • Mr.Math
Yep, so they are NOT coprime.
Mr.Math
  • Mr.Math
As the question asked.
asnaseer
  • asnaseer
note to self: must learn to read the question properly!
asnaseer
  • asnaseer
then your answer seems to be correct.
Mr.Math
  • Mr.Math
I think so, but that was too easy. Congrats on being a moderator, they couldn't have chosen any better :-)
asnaseer
  • asnaseer
thx Mr.Math - I am very humbled to have been chosen. FFM did say this is quite easy, but you are right - it seems TOO easy :-)
Mr.Math
  • Mr.Math
Plus, he always says that even when it's TOO difficult. So I never trust his judgement. :D
asnaseer
  • asnaseer
he he - yes - I concur. I'll have to mull over this one tomorrow. it's quite late here so I need to get some sleep. bye for now...
Mr.Math
  • Mr.Math
Good night.
anonymous
  • anonymous
Oh well not that easy ;) Does \( K=2 \) or\( K=4 \) works for any co-prime in\( 1\le a_i \le 120 \) ?
anonymous
  • anonymous
Whenever I say that the problem is easy, it means that there exists a very short solution for that problem which may or may not use some well known theorems or results, and if I remember correctly I have posted only one too difficult problem here.
anonymous
  • anonymous
and now I have solved this one, I would say this is a easy problem ;)

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