Question 44
For a positive integer $m \ge 2$, define
$$
A_m := \{ m+1, 3m+2, 5m+3, 7m+4, 9m+5, \dots \}.
$$
(1) Prove that, for any given $m \ge 2$, there exists a positive integer $a$, $1 \le a \lt m$, such that either $2^a \in A_m$ or $2^a + 1 \in A_m$.
(2) Assume that, for some $m \ge 2$, there exist positive integers $a$ and $b$ for which $2^a \in A_m$ and $2^b + 1 \in A_m$. Let $a_0$ and $b_0$ be the smallest such $a$ and $b$, respectively, and find a relation between $a_0$ and $b_0$.
$$
A_m := \{ m+1, 3m+2, 5m+3, 7m+4, 9m+5, \dots \}.
$$
(1) Prove that, for any given $m \ge 2$, there exists a positive integer $a$, $1 \le a \lt m$, such that either $2^a \in A_m$ or $2^a + 1 \in A_m$.
(2) Assume that, for some $m \ge 2$, there exist positive integers $a$ and $b$ for which $2^a \in A_m$ and $2^b + 1 \in A_m$. Let $a_0$ and $b_0$ be the smallest such $a$ and $b$, respectively, and find a relation between $a_0$ and $b_0$.