Question 10
Consider the sequence $a_1,a_2,\ldots$, where
$$a_n=\underbrace{111111\ldots1}_{3^n\text{ digits}},$$
that is, $a_1=\underbrace{111}_{\text{three ones}}$, $a_2=\underbrace{111111111}_{\text{nine ones}}$, $a_3=\underbrace{111\ldots1}_{\text{twenty-seven ones}}$, and so on.
(a) Show that $a_1$ is a multiple of $3$ but not of $9$.
(b) Show that $a_2$ is a multiple of $9$ but not of $27$.
(c) Show that $a_3$ is a multiple of $27$ but not of $81$.
$$a_n=\underbrace{111111\ldots1}_{3^n\text{ digits}},$$
that is, $a_1=\underbrace{111}_{\text{three ones}}$, $a_2=\underbrace{111111111}_{\text{nine ones}}$, $a_3=\underbrace{111\ldots1}_{\text{twenty-seven ones}}$, and so on.
(a) Show that $a_1$ is a multiple of $3$ but not of $9$.
(b) Show that $a_2$ is a multiple of $9$ but not of $27$.
(c) Show that $a_3$ is a multiple of $27$ but not of $81$.