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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 $7$?

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 $7$.

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

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