Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$ and $n$ has $k$ prime factors, and $n^*$ is the product of the first $k$ prime factors?
I understand that $\frac{1}{n^*} > \frac{1}{n}$ but how is it that $\phi(n^*) >= \phi(n)$?
Suppose
$$\begin{align*}&n=\prod_{k=1}^m p_k^{a_k}\\{}\\ &n^*=\prod_{k=1}^mp_k\end{align*}\;\;\;\implies \phi(n^*)=n^*\prod_{k=1}^m\left(1-\frac1{p_k}\right)\;,\;\;\phi(n)=n\prod_{k=1}^m\left(1-\frac1{p_k}\right)\implies$$
$$\frac{\phi(n^*)}{n^*}<\frac{\phi(n)}n\iff\prod_{k=1}^m\left(1-\frac1{p_k}\right)<\prod_{k=1}^m\left(1-\frac1{p_k}\right)$$
and the last inequality is clearly impossible