a) Let S ⊂ Z+. What is the smallest value for | S\ that guarantees the existence of two elements x, y ∈ S where x and y have the same remainder upon division by 1000?
b) What is the smallest value of n such that whenever S ⊂ Z+ and | S'| = n, then there exist three elements x, y, z ∈ S where all three have the same remainder upon division by 1000?
c) Write a statement that generalizes the results of parts (a) and (b) and Example 5.42.