## anonymous 4 years ago 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 ?

1. anonymous

I should go to bed now, Enjoy guys :)

2. asnaseer

good night FFM

3. asnaseer

and thanks for the post

4. anonymous

You are always welcome :)

5. Mr.Math

I guess k=4.

6. asnaseer

isn't it k=2 --> {1,2} are 1 and 2 considered to be co prime?

7. Mr.Math

2 is not composite.

8. asnaseer

ah! - of course - thx for pointing that out Mr.Math

9. Mr.Math

Neither is 1.

10. asnaseer

I thought 1 is coprime?

11. asnaseer

but not composite

12. 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.

13. Mr.Math

14. asnaseer

but 2 divides 4 and 6?

15. Mr.Math

Yep, so they are NOT coprime.

16. Mr.Math

17. asnaseer

note to self: must learn to read the question properly!

18. asnaseer

19. Mr.Math

I think so, but that was too easy. Congrats on being a moderator, they couldn't have chosen any better :-)

20. 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 :-)

21. Mr.Math

Plus, he always says that even when it's TOO difficult. So I never trust his judgement. :D

22. 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...

23. Mr.Math

Good night.

24. anonymous

Oh well not that easy ;) Does \( K=2 \) or\( K=4 \) works for any co-prime in\( 1\le a_i \le 120 \) ?

25. 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.

26. anonymous

and now I have solved this one, I would say this is a easy problem ;)