Question 20
Call *pure* any positive integer $n$ that does not occur in any integer sequence $c_0, c_1, c_2, \dots$, where $0 \lt c_0 \lt n$ and
$$
c_i = \begin{cases} \frac{1}{2}c_{i-1} & \text{if } c_{i-1} \text{ is even,} \\ 3c_{i-1} - 1 & \text{if } c_{i-1} \text{ is odd,} \end{cases}
$$
for every $i \ge 1$. (For instance, $10$ is not pure since it occurs in the sequence $5, 14, 7, 20, 10, \ldots$ ...)
a) Is every positive multiple of $3$ pure?
b) Prove that if an integer $n \gt 1$ is pure but not divisible by $3$, then $n + 1$ is divisible by $6$.
(Seniors.)
$$
c_i = \begin{cases} \frac{1}{2}c_{i-1} & \text{if } c_{i-1} \text{ is even,} \\ 3c_{i-1} - 1 & \text{if } c_{i-1} \text{ is odd,} \end{cases}
$$
for every $i \ge 1$. (For instance, $10$ is not pure since it occurs in the sequence $5, 14, 7, 20, 10, \ldots$ ...)
a) Is every positive multiple of $3$ pure?
b) Prove that if an integer $n \gt 1$ is pure but not divisible by $3$, then $n + 1$ is divisible by $6$.
(Seniors.)