Narayanganj Higher Secondary 2014 P9

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samiul_samin
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Narayanganj Higher Secondary 2014 P9

Unread post by samiul_samin » Mon Feb 18, 2019 7:58 pm

At least how many numbers are needed to be taken to be sure that there are at least $11$ numbers among these numbers where the difference between any two is divisible by $7$?

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: Narayanganj Higher Secondary 2014 P9

Unread post by Ragib Farhat Hasan » Thu Oct 31, 2019 8:29 pm

At least $8$ numbers are needed to be taken to ensure that the difference between at least two numbers is divisible by $7$.

Now, if we take one more number, the difference between this and one of the "original" $8$ numbers is divisible by $7$.

Going on, we can see that at least $17$ numbers are required to be taken to ensure that there are at least $11$ numbers among them so that the difference between any two of those $11$ numbers is divisible by $7$.

Ragib Farhat Hasan
Posts: 62
Joined: Sun Mar 30, 2014 10:40 pm

Re: Narayanganj Higher Secondary 2014 P9

Unread post by Ragib Farhat Hasan » Thu Oct 31, 2019 8:30 pm

There is a much more formal solution to this problem using Modular Arithmetic.

But I'm too lazy to type that long a solution! :D

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