ILL 1971 Problem 2

Let us denote by $s(n)= \sum_{d|n} d$ the sum of divisors of a positive integer $n$ ( $1$ and $n$ included). If $n$ has at most $5$ distinct prime divisors, prove that $s(n) < \frac{77}{16} n.$ Also prove that there exists a natural number $n$ for which $s(n) < \frac{76}{16} n$ holds.