Correct Answer is 7
If you have a set of 8 positive integers, and you want to ensure that there exists a pair of numbers in the set with the same remainder when divided by a certain number, you can use the Pigeonhole Principle.
The Pigeonhole Principle states that if you distribute n items into m containers and n>m, then at least one container must contain more than one item.
In this context, the "pigeonholes" are the possible remainders when dividing by a specific number, and the "pigeons" are the 8 positive integers. If you have 8 integers and you want to ensure that there is a pair with the same remainder when divided by a certain number, you can use the Pigeonhole Principle with m=7 (since there are 8 possible remainders when dividing by a number from 0 to 7).
So, there will always be at least one pair of numbers in the set that have the same remainder when divided by the chosen number.
