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Mathematical Puzzles
lunch boxes

Fri Mar 04, 2011 5:03 am  by tartle

‎1 day the principal summoned 1000 students in the quad.. there are 1000 aligned lunch box in the quad..
The principal asked Student number 1 to open every single lunch box, from 1-1000. and asked Student number 2 to come in and close every even lunch box (2,4,6 etc until 1000).
and asked Student number 3 to go to every 3rd lunch box and opens it f its close, and closes it if it s open.
and the 4th student will go to every 4th lunch box and opens it if its close, closes it if its open..
and continues until the 1000th student..
Q: after the 1000th student.. what do u think is the exact number of open lunch-boxes? why?
Sat Mar 19, 2011 11:06 pm  by ChibiHoshi

All integers can be factored. Prime numbers only have the two factors, 1 and itself, so are even, the only number with odd factors will be the perfect squares; 1,3,9 for 9, 1,2,4,8,16 for 16, etc. The rest are always in pairs: 1,2,4,8 for 8 and 1,3,9,11,33,99 for 99 for example.

All the boxes that are perfect squares will be open as they are the only ones odd, all the rest will be closed as they have even factors.

961 is the last perfect square in the sequence
the next is 1024
961 is the square of 31 so 31 boxes will be open
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