Description:There was 100 students and 1000 closed doors. The 1st student opens all 1000 doors, the 2nd student closes doors 2,4,6,8,10, etc., the 3rd student opens doors closed and
closes doors opened on doors 3,6,9,12,15,etc.
after going all of 100 students how many doors are still open?
Answer is 524
How to get it? what are these numbers? What are the properties of these numbers? Do you want to know it?
1) 1 st consider first 100 doors.
Only 10 doors are open after passing all students.
door number students number
1 (1)
4 (1, 2, 4)
9 (1, 3, 9)
16 (1, 2, 4, 8, 16)
25 (1, 5, 25)
36 (1, 2, 3, 4, 6, 9, 12, 18, 36)
49 (1, 7, 49)
64 (1, 2, 4, 8, 16, 32, 64)
81 (1, 3, 9, 27, 81)
100 (1, 2, 4, 5, 10, 20, 25, 50, 100)
These are Square Number
49 doors are remains open after passing all students.
Apply above method to get all such numbers.
6) Consider doors 501 to 600
53
7) Consider doors 601 to 700
54
8) Consider doors 701 to 800
53
9) Consider doors 801 to 900
48
10) Consider doors 901 to 1000
53
after going all of 100 students how many doors are still open?
Answer is 524
How to get it? what are these numbers? What are the properties of these numbers? Do you want to know it?
1) 1 st consider first 100 doors.
Only 10 doors are open after passing all students.
door number students number
1 (1)
4 (1, 2, 4)
9 (1, 3, 9)
16 (1, 2, 4, 8, 16)
25 (1, 5, 25)
36 (1, 2, 3, 4, 6, 9, 12, 18, 36)
49 (1, 7, 49)
64 (1, 2, 4, 8, 16, 32, 64)
81 (1, 3, 9, 27, 81)
100 (1, 2, 4, 5, 10, 20, 25, 50, 100)
These are Square Number
- Student 1 can open all the doors.
- The Students whose number is same as door's number they are close the doors.
- If the door number is prime it can't open by other students.Because it can't have other divisors.
- If the door number is composite number and it is not a square number then these doors are closed after passing all students. Because door number = M x N. If M is open the door then N is closed it. Let door number is 10. 10= 2 x 5, 2 open the door and 5 close that door. Let door number is 20 , 20 = 2 x 10,4 x 5. Here 2 open the door and 10 close that door, 4 open the door and 5 close that door.
- If the door number is composite number and it is a square number then these doors are opened after passing all students. Because one pair of divisor is in the form of N x N. Let door number is 25. 25 = 5 x 5. Here student 5 open the door. Let door number is 16. 16 = 2 x 8, 4 x 4. Here student 2 open the door student 8 close it and student 4 open it.
Only 10 doors are open after passing all students.
2) Consider doors 101 to 200
All factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 96 doors are remains open after passing all students. Door number 121,144,169 and 196(square numbers) are remains closed.
96 doors are remains open after passing all students.
3) Consider doors 201 to 300
->If door's number is odd, then all factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 48 doors are remains open after passing all students. Door number 225 and 289(square numbers) are remains closed.
->If door's number is even, one of the factors of door's number is grater than 101. Here student 1open all doors, and student 2 closed all doors. If another student (number M !=2) can open the door there exist a student (number N!=2) can close it. But in the case of Square Numbers M=N therefore door remains open. There was is only 1 square number 256. Only 1 door is remains open after passing all students.All factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 96 doors are remains open after passing all students. Door number 121,144,169 and 196(square numbers) are remains closed.
96 doors are remains open after passing all students.
3) Consider doors 201 to 300
->If door's number is odd, then all factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 48 doors are remains open after passing all students. Door number 225 and 289(square numbers) are remains closed.
49 doors are remains open after passing all students.
4) Consider doors 301 to 400
->If door's number is not a multiple of 2 or 3, then all factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 32 doors are remains open after passing all students. Door number 361(square number) is remains closed.
->If door's number is multiple of 6, two of the factors of door's number is grater than 101(2 x N and 3 x N). Here student 1open all doors, student 2 closed all doors, and student 3 open all doors. If another student (number M !=2 or 3) can close the door there exist a student (number N!=2 or 3) can open it. But in the case of Square Numbers M=N therefore door remains close. There is only 1 square number 324. 15 door is remains open after passing all students.
->If door's number is multiple of 2 and not multiple of 3, one of the factors of door's number is grater than 101(2 x N). Here student 1 open all doors, and student 2 closed all doors. If another student (number M !=2) can open the door there exist a student (number N!=2) can close it. But in the case of Square Numbers M=N therefore door remains open. There is only 1 square number 400. Only 1 door is remains open after passing all students.
->If door's number is multiple of 3 and not multiple of 2, one of the factors of door's number is grater than 101(3 x N). Here student 1 open all doors, and student 3 closed all doors. If another student (number M !=3) can open the door there exist a student (number N!=3) can close it. But in the case of Square Numbers M=N therefore door remains open. There is no such square number. all doors are remains close after passing all students.
48 doors are remains open after passing all students.
->If door's number is not a multiple of 2 or 3, then all factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. 32 doors are remains open after passing all students. Door number 361(square number) is remains closed.
->If door's number is multiple of 6, two of the factors of door's number is grater than 101(2 x N and 3 x N). Here student 1open all doors, student 2 closed all doors, and student 3 open all doors. If another student (number M !=2 or 3) can close the door there exist a student (number N!=2 or 3) can open it. But in the case of Square Numbers M=N therefore door remains close. There is only 1 square number 324. 15 door is remains open after passing all students.
->If door's number is multiple of 2 and not multiple of 3, one of the factors of door's number is grater than 101(2 x N). Here student 1 open all doors, and student 2 closed all doors. If another student (number M !=2) can open the door there exist a student (number N!=2) can close it. But in the case of Square Numbers M=N therefore door remains open. There is only 1 square number 400. Only 1 door is remains open after passing all students.
->If door's number is multiple of 3 and not multiple of 2, one of the factors of door's number is grater than 101(3 x N). Here student 1 open all doors, and student 3 closed all doors. If another student (number M !=3) can open the door there exist a student (number N!=3) can close it. But in the case of Square Numbers M=N therefore door remains open. There is no such square number. all doors are remains close after passing all students.
48 doors are remains open after passing all students.
5) Consider doors 401 to 500
->If door's number is not a multiple of 2 or 3 or 4, then all factors of door's number are less than 101. Here student 1 open all doors, and if another student (number M) can close the door there exist a student (number N) can open it. But in the case of Square Numbers M=N therefore door remains closed. There is no such square number. 34 doors are remains open after passing all students.
->If door's number is multiple of 2 and 3 not multiple of 4, two of the factors of door's number is grater than 101(2 x N and 3 x N). Here student 1 open all doors, student 2 closed all doors, and student 3 open all doors. If another student (number M !=2 or 3) can close the door there exist a student (number N!=2 or 3) can open it. But in the case of Square Numbers M=N therefore door remains close. There is no such square number. 9 door is remains open after passing all students.
->If door's number is multiple of 2 and 4 not multiple of 3, two of the factors of door's number is grater than 101(2 x N and 4 x N). Here student 1 open all doors, student 2 closed all doors, and student 4 open all doors. If another student (number M !=2 or 3) can close the door there exist a student (number N!=2 or 4) can open it. But in the case of Square Numbers M=N therefore door remains close. There is only 1 square number 484. 16 door is remains open after passing all students.
->If door's number is multiple of 2 and not multiple of 3 or 4, one of the factors of door's number is grater than 101(2 x N). Here student 1 open all doors, and student 2 closed all doors. If another student (number M !=2) can open the door there exist a student (number N!=2) can close it. But in the case of Square Numbers M=N therefore door remains open. There is no such square number.
->If door's number is multiple of 3 and not multiple of 2, one of the factors of door's number is grater than 101(3 x N). Here student 1 open all doors, and student 3 closed all doors. If another student (number M !=3) can open the door there exist a student (number N!=3) can close it. But in the case of Square Numbers M=N therefore door remains open. Door number 441(square number) is remains open. 1 door remains open after passing all students.
60 doors are remains open after passing all students.
Apply above method to get all such numbers.
6) Consider doors 501 to 600
53
7) Consider doors 601 to 700
54
8) Consider doors 701 to 800
53
9) Consider doors 801 to 900
48
10) Consider doors 901 to 1000
53
What are the properties of these numbers?
1. Prime test.
All prime number between 100 to 1000 are include in this set.
2. Square number.
All square number less than 100 include in this set.
3. Properties of single set(set contain 100 elements):
- >101 to 200 hset(), sset(121,144,169,196)
- >201 to 300 hset(2), sset(225,256,289)
- >301 to 400 hset(2,3), sset(324,361,400)
- >401 to 500 hset(2,3,4), sset(441,484)
- >501 to 600 hset(2,3,4,5), sset(529,576)
- >601 to 700 hset(2,3,4,5,6), sset(625,676)
- >701 to 800 hset(2,3,4,5,6,7), sset(729,784)
- >801 to 900 hset(2,3,4,5,6,7,8), sset(841)
- >901 to 1000 hset(2,3,4,5,6,7,8,9), sset(900,961)
Let M be the door's number>100 ,
List of all open door with students who open or close that door, And Python code to find these numbers is given below.
1. Prime test.
All prime number between 100 to 1000 are include in this set.
2. Square number.
All square number less than 100 include in this set.
3. Properties of single set(set contain 100 elements):
- >101 to 200 hset(), sset(121,144,169,196)
- >201 to 300 hset(2), sset(225,256,289)
- >301 to 400 hset(2,3), sset(324,361,400)
- >401 to 500 hset(2,3,4), sset(441,484)
- >501 to 600 hset(2,3,4,5), sset(529,576)
- >601 to 700 hset(2,3,4,5,6), sset(625,676)
- >701 to 800 hset(2,3,4,5,6,7), sset(729,784)
- >801 to 900 hset(2,3,4,5,6,7,8), sset(841)
- >901 to 1000 hset(2,3,4,5,6,7,8,9), sset(900,961)
Let M be the door's number>100 ,
if M is open after passing all students,
then conditions:
1) Odd number of elements in hset is factor of M and M is in sset.
OR
2) Zero or even number of elements in hset is factor of M and M is not in sset.
List of all open door with students who open or close that door, And Python code to find these numbers is given below.
# To store status of each door 1 stands for closed -1 stands for opened.
status=[]
#To store list of student who open or close each door.
door=[]
for d in range(1,1001):
#Initially status of all door is closed.
status.append(1)
#Initially none of student open or closed the door.
door.append(())
for student in range(1,101):
k=student
while(k<=1000):
#change status of the door
status[k-1] = -(status[k-1])
#Add the student to the list
door[k-1]=door[k-1]+(student,)
# Find multiplication of student's number
k +=student
#To count opened door after passing allstudent
count=0
ans = open("door.txt","w")
for d in range(1,1001):
if(status[d-1] == -1):
#Filter the open doors
#Increase count of open door by 1
count +=1
#Print the door number and students who open or closed the door.
print >>ans,d,door[d-1]
#print the total number of open doors
print count
