ISL 2005 N5

Denote by $d(n)$ the number of divisors of the positive integer $n$ . A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$ . Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$ .

(a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$ .

(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.