For a given integer $k$, which of the following are false?

Question

For a given integer $k$, which of the following are false?

$(1)$ If $k($mod $72)$ is a unit in $\mathbb{Z}_{72}$, then $k($ mod $9)$ is a unit in $\mathbb{Z}_9$

$(2)$ If $k($mod $72)$ is a unit in $\mathbb{Z}_{72}$, then $k($ mod $8)$ is a unit in $\mathbb{Z}_8$

$(3)$ If $k($mod $8)$ is a unit in $\mathbb{Z}_{8}$, then $k($ mod $72)$ is a unit in $\mathbb{Z}_{72}$

$(4)$ If $k($mod $9)$ is a unit in $\mathbb{Z}_{9}$, then $k($ mod $72)$ is a unit in $\mathbb{Z}_{72}$

I am confused to proceed in any direction .Please help.

Thanks for your time and support.

Answer 1

$k$ is a unit modulo $n$ iff there exists a $b$ such that $kb \equiv 1 \pmod{n}$. This implies that $$n | (kb-1)$$ Now suppose $d | n$. Then by transitive property of divisibility, we can claim that $d | (kb-1)$. So $kb \equiv 1 \pmod{d}$ as well. This means $k$ is a unit modulo $d$ as well.

For the converse, let us consider $3$. It is a unit in $\Bbb{Z}_8$ because $3^2 \equiv 1 \pmod{8}$. Ask yourself: is $3$ a unit in $\Bbb{Z}_{72}$ or not?