Prove the following property for Euler's totient function I have been asked to prove that $\phi(n) > \dfrac{9n}{50}$ for all $n$ that have at most seven prime factors. I'm trying to think of how this relates to any theorems/ properties I have learnt regarding the totient function, but I have drawn a blank. Could anyone help to explain how the property holds?
 A: The important part is to know the first few primes, and the formula for the totient function.

 $$n = \Pi_{k=0}^{m}p_k^{\alpha_k} \\ \phi (n)=n\Pi_{k=0}^{m}(1-\frac1{p_k})$$


 There are at most 7 prime factors, and this can give us a lower bound. The terms in the RHS product are fractions smaller than 1, so increasing the number of terms to 7 would minimize the product. In order to make the 7 fractions as small as possible, the primes need to also be as small as possible. Hence choosing the first 7 primes would minimize the product.
 $$ \phi (n)=n\Pi_{k=0}^{m}(1-\frac1{p_k})\geq n\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{11}\right)\left(1-\frac{1}{13}\right)\left(1-\frac{1}{17}\right)$$


 Expand the fractions, and if you will forgive the usage of a calculator, this does prove the statement.
 $$ \phi (n)=n\Pi_{k=0}^{m}(1-\frac1{p_k})\geq n\frac{3072}{17017}>\frac9{50}n$$

I imagine that there is a neater proof, but I cannot think of one at the moment.
