Prove that $7\sqrt{5}$ is an irrational number.
Prove that $7\sqrt{5}$ is an irrational number.
Show Hint
Solution and Explanation
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.
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