Question

Prove that a real number 𝑥 is irrational if and only if 5𝑥 is irrational

Answer 1

$Let\,\,x\notin \mathbb{Q} . Suppose\,\,5x\in \mathbb{Q} ,then\,\,x=\frac{5x}{5}\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Let\,\,5x\notin \mathbb{Q} . Suppose\,\,x\in \mathbb{Q} , then\,\,5x=5\cdot x\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Thus\\x\notin \mathbb{Q} \Leftrightarrow 5x\notin \mathbb{Q}$