Problem Statement
Prove that $\sqrt{5}$ is an irrational number.
Verified Solution & Marking Scheme
Assume Rationality
Assume $\sqrt{5} = \frac{a}{b}$ where $a, b$ are coprime integers ($b \neq 0$). Then $5b^2 = a^2$.
Divisibility of a
$5$ divides $a^2 \implies 5$ divides $a$. Let $a = 5c$. Then $5b^2 = (5c)^2 = 25c^2 \implies b^2 = 5c^2$.
Contradiction
Thus 5 divides $b^2 \implies 5$ divides $b$. This contradicts that $a$ and $b$ are coprime. Hence $\sqrt{5}$ is irrational.