Question

Prove that $7\sqrt{5}$ is an irrational number.
 

Hint:

The product of a nonzero rational number and an irrational number is always irrational.

Answer 1

Step 1: Assume the contrary.
Suppose $7\sqrt{5}$ is rational. Then it can be expressed as \[ 7\sqrt{5}=\frac{m}{n}, m,n\in\mathbb{Z},\ n\neq 0,\ \gcd(m,n)=1. \]

Step 2: Divide by the rational number $7$.
Since $7$ is nonzero and rational, \[ \sqrt{5}=\frac{m}{7n}, \] which is rational.

Step 3: Contradiction.
It is already known that $\sqrt{5}$ is irrational. Hence, our assumption is false. Conclusion:
Therefore, $7\sqrt{5}$ is irrational.