## Narayanganj Higher Secondary 2014 P9

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samiul_samin
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Joined: Sat Dec 09, 2017 1:32 pm

### Narayanganj Higher Secondary 2014 P9

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

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

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

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