There are 100 switches labeled from 1 to 100 corresponding to each of the bulbs. 10 persons are asked to report the number of defective bulbs they had found, such that the 1st person is allowed to check all the bulbs while the 2nd person is allowed to check bulbs corresponding to switches labeled 2,4,...,100 and so on, and the 10th person is allowed to check bulbs corresponding to switches labeled 10,20,...,100.

Each person reported the same number of defective bulbs. Also note that the number of defective bulbs can increase or remain the same between two consecutive testings and there is at least one defective bulb. Find the following just after the 10th person finished his test.

1) What can be the minimum number of defective bulbs? What are the defective bulbs in this case?

2) What can be the maximum number of defective bulbs? What can be the maximum number of bulbs that can be found to be defective?

The maximum possible number of bulbs reported is 86. This can be seen from the fact that there are 22 numbers which can be reported only by person 1 (1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) And there are 12 numbers which can be reported only be persons 1 or 2 (2, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94) No matter what numbers are reported, there must be at least 14 unreported.

86 defective bulbs can be reported when all 10 people report 10 defective bulbs:

P1:1,11,13,17,19,23,29,31,37,41

P2:2,22,26,34,38,46,58,62,74,82

P3:3,33,39,51,57,69,84,87,93,96

P4:4,12,28,44,52,68,76,84,92,96

P5:5,15,25,35,45,55,65,75,85,95

P6:6,12,24,30,42,48,66,78,84,96

P7:7,14,21,35,42,49,56,84,91,98

P8:8,16,24,32,40,48,56,64,88,96

P9:9,18,27,36,45,54,63,72,81,99

P10:10,20,30,40,50,60,70,80,90,100