Give an example of a natural number $n$ such that $\phi(n) < \frac{n}{3}$.
Give reasons for your choice.
Proposition (5.9) tell us that
$$ \phi(n) = n \times \biggl(1- \frac{1}{p_1}\biggr) \times \biggl(1- \frac{1}{p_2}\biggr) \times \ldots \times \biggl(1- \frac{1}{p_r}\biggr) $$
where $p_i$ are distinct primes.
We saw in the previous exercise that the inclusion of 2 and 3 as primes leads to $\phi(n) =\frac{n}{3}$.
The inclusion of any other prime reduces the size of the product because $(1-\frac{1}{p})<1$ for any prime.
If we choose to include the prime 5, then for any $n=2^a3^b5^c$ we have $\phi(n)<\frac{n}{3}$, for natural numbers $a,b,c$.
The simplest example is $n=2 \times 3 \times 5 = 30$.
As a check $\phi(30) = (2-1) \times (3-1) \times (5-1) = 8$, which is indeed less than a third of $30$.