Lemma nice criterion for non-intersecting integers

Let $n$ be an integer with at least one prime divisor, and let $p$ be the smallest prime that divides $n$. Let non-intersecting mean that the relevant congruence classes with moduli given by the divisors of $n$ do not form an intersecting covering system. If

$$\frac{n}{p}$$

has fewer than $p$ distinct prime divisors, then $n$ is non-intersecting.

Lemma nice criterion. If $p$ is the smallest prime divisor of $n$ and $n/p$ has fewer than $p$ distinct prime divisors, then $n$ is non-intersecting.

This is the sufficient condition attributed in the surrounding discussion to Lemma; the paper then conjectures that the same condition is also necessary. No resolution is supplied in the given text.