## shubhamsrg 3 years ago In a jail with 100 rooms, all locked initially, 100 rioters break in and disturb the rooms in the following way. First one stops at all rooms and opens them all. Second rioters stops at rooms numbered 2, 4, 6, . . . and locks open rooms, and leaving the other rooms as they were. The third rioter stops at rooms numbered 3, 6, 9, . . . and again opens a locked room and locks an open room, leaving others undisturbed. And this process continues. After all the 100 rioters have left which rooms would be open?

1. shubhamsrg

i dont know! :P any good reasons ?

2. lgbasallote

the ones that will be open are the squares (i.e. 4, 9, 16, 25, etc)

3. lgbasallote

the ones that will be closed are the not squares

4. sauravshakya

@lgbasallote I think room 1 will be surely opened..

5. lgbasallote

yes it will

6. lgbasallote

1 is a perfect square isnt it

7. lgbasallote

1 opens 4 2 closes 4 4 opens 4

8. Yahoo!

100 th room

9. lgbasallote

for 9 1 opens 9 3 closes 9 9 opens 9

10. lgbasallote

for 16 1 opens 16 2 closes 16 4 opens 16 8 closes 16 16 opens 16

11. lgbasallote

for 25 1 opens 25 5 closes 25 25 opens 25

12. lgbasallote

for 36 1 opens 36 3 closes 36 6 opens 36 12 closes 36 36 opens 36

13. lgbasallote

for 49 1 opens 49 7 closes 49 49 opens 49

14. sauravshakya

I guess the number which has odd number of factors will be open

15. shubhamsrg

in 36 you missed some in between but result remains open,,hmm i see maybe you're correct..

16. sauravshakya

And the room number which has even number of factors will be closed

17. lgbasallote

for 64 1 opens 64 2 closes 64 4 opens 64 8 closes 64 16 opens 64 32 closes 64 64 opens 64

18. lgbasallote

what comesafter 64 again?

19. shubhamsrg

81 1o 3c 9o 27c 81o

20. lgbasallote

yes that

21. shubhamsrg

finally 100 1o 2c 4o 5c 10o 20c 25o 50c 100o nice.. thanks..

22. lgbasallote

welcome

23. shubhamsrg

can there be some reason why its always sq. no ? or its just a pattern ? ! ?

24. lgbasallote

it's a tedious proof... lol

25. lgbasallote

i just know the trial and error proof of it

26. sauravshakya

I guess only sq. no have odd number of factors

27. shubhamsrg

aha..yes it seems so..lol..might make up another good ques that only sq no. have odd no. of factors! :D

28. shubhamsrg

damn i closed the quess,,wait i shall tag,,

29. shubhamsrg

@mukushla @eliassaab

30. sauravshakya

Oh..... got it....... I think I can prove it

31. sauravshakya

IF its not an sq. number.... for example 12 then the factors are 1,2,3,4,6,12

32. sauravshakya

right?

33. shubhamsrg

yes..

34. sauravshakya

|dw:1345558048314:dw|Now,

35. sauravshakya

Now, lets look a sq. number ... for example 25 its factors are 1,5,5,25|dw:1345558167578:dw|

36. shubhamsrg

hmm..so every no. will have even no. of factors if we count same no. twice..really ? that quite amazing!

37. sauravshakya

If it is not a sq. number than there will always be a unique number for its factor.... so that their product will be that number

38. sauravshakya

TO make it more clear......|dw:1345558421778:dw|

39. sauravshakya

and sum of even numbers is always a even number

40. shubhamsrg

cool..i get it! nice.. :)

41. shubhamsrg

now lets return to my ques,,why only sq. nos should be there?

42. sauravshakya

To...... write and make other understand is a very difficult......

43. sauravshakya

without speaking

44. sauravshakya

Am i right?

45. shubhamsrg

lol..you did a pretty good job sir.,thanks..

46. sauravshakya

I mean it is easy for u guys........... but for others its really hard

47. shubhamsrg

am not very good at the subject sir,,so explaining to me always takes some effort! see,,am a bad student! hmm.. nevermind,,back to the ques please! why only sq. nos should be there?

48. mukushla

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