TITLE: Condition on two irrational numbers
QUESTION [5 upvotes]: Consider two irrational numbers $a$ and $b$. Are there well known sufficent conditions on the relation between $a$ and $b$ that would allow me to conclude that there is $c \in \mathbb{R}$, $c \neq 0$, such that both $ca$ and $cb$ are rational?

REPLY [8 votes]: If $a=qb$ for some $q\in\mathbb{Q}$ then the condition clearly holds. For the other implication, say there exists $0\neq c\in\mathbb{R}$ with $ca$ and $cb$ are rational. Then let $q=ca/cb$. Then $a=qb$. 
Therefore there exists a $0\neq c\in\mathbb{R}$ with $ca$ and $cb$ rational iff $a=qb$ for some $q\in\mathbb{Q}$.