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There are 20 doors marked with numbers 1 to 20. And there are 20 indiv

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There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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New post 24 Sep 2015, 09:26
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Question Stats:

56% (02:46) correct 44% (02:13) wrong based on 116 sessions

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There are 20 doors marked with numbers 1 to 20. And there are 20 individuals marked 1 to 20.

An operation on a door is defined as changing the status of the door from open to closed or vice versa. All doors are closed to start with.

Now one at a time one randomly picked individual goes and operates the doors. The individual however operates only those doors which are a multiple of the number he/she is carrying. For e.g. individual marked with number 5 operates the doors marked with the following numbers: 5, 10, 15 and 20.

If every individual in the group get one turn then how many doors are open at the end?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 6
[Reveal] Spoiler: OA

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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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New post 24 Sep 2015, 09:37
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rajarshee wrote:
There are 20 doors marked with numbers 1 to 20. And there are 20 individuals market with number 1 to 20.

An operation on a door is defined as changing the status of the door from open to closed or vice versa. All doors are closed to start with.

Now one at a time one randomly picked individual goes and operates the doors. The individual however operates only those doors which are a multiple of the number he/she is carrying. For e.g. individual marked with number 5 operates the doors marked with the following numbers: 5, 10, 15 and 20.

If every individual in the group get one turn then how many doors are open at the end?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 6


the Q has to do with the properties of number..
when the door is closed , it requires odd number of people to operate to be opened...
only perfect squares have odd number of factors ..
so ans is door no 1,4,9,16.. 4 doors
D
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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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New post 24 Sep 2015, 10:06
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Hi rajarshee,

This question has a 'visual component' to it that you can use to 'track' when each door is opened or closed. While the pattern that chetan2u describes is what defines the correct answer, most people can't make that deduction immediately; as such, you have to do enough work to PROVE that that pattern exists.

Rather than work through all 20 people and all 20 doors, I'm going to work through the first several so that we can define the pattern involved...

Remember: All the doors start off CLOSED...
Door 1: Only Person 1 touches this door. So it IS OPEN at the end.
Door 2: Person 1 and Person 2 touch this door. So it is closed at the end.
Door 3: Person 1 and Person 3 touch this door. So it is closed at the end.
Door 4: Person 1, 2 and 4 touch this door. So it IS OPEN at the end.

Now, stop and look at the work that we've done so far... Which doors do we know for sure will be open? Door 1 and Door 4. What do those two numbers have in common? They're both PERFECT SQUARES..... Let's see if that pattern continues...

Door 5: Person 1 and 5 touch this door. CLOSED.
Door 6: Person 1, 2, 3 and 6 touch this door. CLOSED.
Door 7: Person 1 and 7. CLOSED.
Door 8: Person 1, 2, 4 and 8. CLOSED
Door 9: Person 1, 3 and 9. OPEN.

Notice how the next door that we know will be open in the end is Door 9. It is ALSO a PERFECT SQUARE. Given the work we've done so far, this MUST be the pattern, so we're ultimately looking for the number of perfect squares from 1 to 20. They are 1, 4, 9 and 16.

Final Answer:
[Reveal] Spoiler:
D


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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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New post 24 Sep 2015, 10:16
rajarshee wrote:
There are 20 doors marked with numbers 1 to 20. And there are 20 individuals market with number 1 to 20.

An operation on a door is defined as changing the status of the door from open to closed or vice versa. All doors are closed to start with.

Now one at a time one randomly picked individual goes and operates the doors. The individual however operates only those doors which are a multiple of the number he/she is carrying. For e.g. individual marked with number 5 operates the doors marked with the following numbers: 5, 10, 15 and 20.

If every individual in the group get one turn then how many doors are open at the end?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 6


Let us start with 1.
When 1 goes, all the doors are open since every number is a multiple of 1.
Also 1 will remain opened since 1 is not a multiple of any number.

Prime numbers can only be operated by people with prime numbers.
So 2,3,5,7,11,13,17,19 will be closed when people with respective numbers perform operations on them.
Our numbers left are 4,6,8,9,10,12,14,15,16,18,20

From these numbers, only those numbers will remain opened on which even number of operations are performed including the operation performed by number 1.
This is possible only for perfect squares which are 4,9 and 16.
So our numbers are 1,4,9 and 16.

Answer:- D

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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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New post 23 Oct 2016, 05:53
rajarshee wrote:
There are 20 doors marked with numbers 1 to 20. And there are 20 individuals marked 1 to 20.

An operation on a door is defined as changing the status of the door from open to closed or vice versa. All doors are closed to start with.

Now one at a time one randomly picked individual goes and operates the doors. The individual however operates only those doors which are a multiple of the number he/she is carrying. For e.g. individual marked with number 5 operates the doors marked with the following numbers: 5, 10, 15 and 20.

If every individual in the group get one turn then how many doors are open at the end?

(A) 0
(B) 1
(C) 2
(D) 4
(E) 6


Similar question to practice: a-high-school-has-a-strange-principal-on-the-first-day-he-has-his-st-101625.html
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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv [#permalink]

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Re: There are 20 doors marked with numbers 1 to 20. And there are 20 indiv   [#permalink] 20 Nov 2017, 23:02
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