Can you please help with Question 5 on the 2008 MAT?
Certainly. The nth student (Sn) changes the state of every nth locker - i.e. the multiples of n. If changed an odd number of times, the locker is open -- if even then closed.
i. how many closed after 3rd student? if closed then the locker is either left open by S2 and closed by S3 or closed by S2 and left closed by S3. i.e [ not(2 | n) & (3 | n) ] or [ not(3 | n) & (2 | n) ].
The first 6 lockers are OCCCOO (this pattern repeats). 1000=1666+4 so there are 1663+3=501 closed lockers. (1663 C's from the 166 cycles, +3 from the incompleted cycle).
ii. After S3 the first 12 lockers are OCCC OOOC CCOO. S4 changes state of 4,8 and 12, leaving OCCO OOOO CCOC. 1000=8312+4 so there are 835+2=417
iii. 100=2^25^2 is changed once for every factor. There are 33=9 factors (number of 2's in factor =0,1,2; of 5's =0,1,2). 9 is odd so locker open.
iv. We need to find number of factors of 1000 = 2^35^3 which are less than or equal to 100. Consider systematically number of 2's and 5's in factor.
1,2,4,8 ; 5,10,20,40 ; 25,50,100 (after this factor too large). So 11 factors (odd) so locker open.
Bonus questions:
- Consider infinite closed lockers. What proportion will be open after infinite students pass?
