Note on squarefree integers through a set theoretical property

Abstract

In this paper, we show a property of set theory, that in number theory has the following consequence: if $a_{1} < a_{2} < \ldots < a_{n}$ are squarefree integers, then the number of distinct ratios $ a_{i}/(a_{i},a_{j})$ is greater than or equal to n, where$(a_{i},a_{j})$ denotes the greatest common divisor of $a_{i}$ and $a_{j}$.