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Partition of N into two subsets

Source: India TST 2016 Day 4 Problem 3

July 22, 2016
number theorysetprime numbers

Problem Statement

Let N\mathbb N denote the set of all natural numbers. Show that there exists two nonempty subsets AA and BB of N\mathbb N such that
[*] A∩B={1};A\cap B=\{1\}; [*] every number in N\mathbb N can be expressed as the product of a number in AA and a number in BB; [*] each prime number is a divisor of some number in AA and also some number in BB; [*] one of the sets AA and BB has the following property: if the numbers in this set are written as x1<x2<x3<⋯x_1<x_2<x_3<\cdots, then for any given positive integer MM there exists k∈Nk\in \mathbb N such that xk+1−xk≥Mx_{k+1}-x_k\ge M. [*] Each set has infinitely many composite numbers.
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