Question 121
A partition of the Set of Natural Numbers is a collection of sets $A_{1}, A_{2}, \ldots, A_{k}$ such that each natural number belongs to exactly one of them. Note that in any partition of the Set of Natural Numbers at least one of these sets is infinite, since otherwise the Set of Natural Numbers would be the union of a finite number of finite sets and would therefore be finite. An example of a partition of the Set of Natural Numbers is to take as $A_{1}$ the set of all even natural numbers and as $A_{2}$ the set of all odd natural numbers. There are many possible partitions, and the next two items are general facts that can be verified in any of these partitions.
a) Explain why, for each fixed positive integer $x$, there is always one of the sets $A_{i}$ containing infinitely many multiples of $x$.
b) By the previous item, given two positive integers $p$ and $q$, there is one set of the partition with infinitely many multiples of $p$ and another set containing infinitely many multiples of $q$. However, these two sets need not be the same. Now show that some set $A_{i}$ always contains infinitely many multiples of every positive integer.
a) Explain why, for each fixed positive integer $x$, there is always one of the sets $A_{i}$ containing infinitely many multiples of $x$.
b) By the previous item, given two positive integers $p$ and $q$, there is one set of the partition with infinitely many multiples of $p$ and another set containing infinitely many multiples of $q$. However, these two sets need not be the same. Now show that some set $A_{i}$ always contains infinitely many multiples of every positive integer.