How can I go about proving the following problem:
Prove that a number $a$ is rational if and only if there exists a positive integer $k$ such that $[ka]=ka$. Prove that a number $a$ is rational if and only if there exists $k$ such that $[k! a]= k! a$
The following chain of reasoning proves both parts of both problems:
If $a$ is rational, then there are integers $p,q$ with $a=p/q$.
If $a=p/q$, then $qa=p$, so $[qa]=[p]=p=qa$.
If $[ka]=ka$, then $ka\in\mathbb{Z}$, then $(ka)(k-1)!\in \mathbb{Z}$, then $[k!a]=k!a$.
If $[k!a]=k!a$, then $k!a=t\in \mathbb{Z}$, then $a=t/k!$, a ratio of two integers, so $a$ is rational.