Suppose S is a set of n + 1 integers. Prove that there exist distinct a, b S such that a - b is a multiple of n. Solution There n distinct remainders modulo n ie 0,1,2,...,n-1 So let there be n boxes each corresponding to one of these remainders We need to put these n+1 integers into these n boxes So two of them must be in the same box ie two of them give the same remainder modulo n Let those two integers be a,b SO,a=b modulo n a-b=0 modulo n ie a-b is a multiple of n HEnce proved.

1. Suppose S is a set of n + 1 integers. Prove that there exist distinct a, b S such that a - b is a
multiple of n.
Solution
There n distinct remainders modulo n ie 0,1,2,...,n-1
So let there be n boxes each corresponding to one of these remainders
We need to put these n+1 integers into these n boxes
So two of them must be in the same box ie two of them give the same remainder modulo n
Let those two integers be a,b
SO,a=b modulo n
a-b=0 modulo n
ie a-b is a multiple of n
HEnce proved