Question 128
A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers.
a) Does there exist a meaningful set which consists of 2019 numbers?
b) Does there exist an infinite meaningful set?
Note: The geometric mean of the non-negative numbers $a_{1}, a_{2}, \ldots, a_{n}$ is defined as $\sqrt[n]{a_{1} a_{2} \cdots a_{n}}$
a) Does there exist a meaningful set which consists of 2019 numbers?
b) Does there exist an infinite meaningful set?
Note: The geometric mean of the non-negative numbers $a_{1}, a_{2}, \ldots, a_{n}$ is defined as $\sqrt[n]{a_{1} a_{2} \cdots a_{n}}$