Question 160
A sequence $\{a_n \mid n = 1, 2, \dots\}$ of positive integers is called an ascending sequence if $a_n \lt a_{n+1}$ and $a_{2n} = 2a_n$ are satisfied for every positive integer $n$.
(1) Suppose $\{a_n\}$ is an ascending sequence. For any prime $p$ greater than $a_1$, show that a multiple of $p$ appears among the terms of the sequence $\{a_n\}$.
(2) Let $p$ be an odd prime. Show that there exists an ascending sequence $\{a_n\}$ for which multiples of $p$ never appear among the terms of $\{a_n\}$.
(1) Suppose $\{a_n\}$ is an ascending sequence. For any prime $p$ greater than $a_1$, show that a multiple of $p$ appears among the terms of the sequence $\{a_n\}$.
(2) Let $p$ be an odd prime. Show that there exists an ascending sequence $\{a_n\}$ for which multiples of $p$ never appear among the terms of $\{a_n\}$.