# How to justify $\phi(n) \ge \sqrt{n}$

If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$?

• $\varphi(2)=1<\sqrt2$. – Brian M. Scott Feb 20 '13 at 2:00
• I'm sure this question has been asked, and answered, here, but I'm not sure how to find it. – Gerry Myerson Feb 20 '13 at 2:01
• $\varphi(6) = 2 < \sqrt{6}$ – davidlowryduda Feb 20 '13 at 2:01
• See, for example, math.stackexchange.com/questions/301837/… --- especially Brian's answer. – Gerry Myerson Feb 20 '13 at 2:08
• – Will Jagy Feb 20 '13 at 2:08

Hint: both $\phi(n)$ and $\sqrt{n}$ are multiplicative, so it suffices to consider prime powers.