We need to minimize $(1-\frac{1}{p_1})(1-\frac{1}{p_2})$ etc.. (using the formula of $\phi(n)$)
$1-\frac{1}{p_1}$ is small if $\frac{1}{p_1}$ is large
The largest that $\frac{1}{p_1}$ can get is $\frac12$ (as $2$ is the smallest prime)
So the smallest $(1–\frac{1}{p_1})$ can get is $\frac12$]
Similarly $(1-\frac{1}{p_2})$ = $\frac23$ (next smallest value)
$\frac12\times\frac23\times\frac45\times\frac67$ is the smallest $\phi(n)$ possible
One number is $210 (2\times 3\times 5\times 7)$. To get the others, increase the power of $2, 3, 5, 7$ such that the number remains less than $1000$
.