2005 IMO Shortlist S06

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.

$d(n)$ 表示正整数 $n$ 的约数数。如果对于所有正整数 $m < n$ 来说 $d(n) > d(m)$，则正整数 $n$ 被称为高度整除。

如果不存在满足 $m < s < n$ 的高整除整数 $s$，则两个 $m < n$ 的高整除整数 $m$ 和 $n$ 被称为连续。

(a) 证明只有有限多对连续的高度整除的
$(a, b)$ 形式的整数，其中 $a\mid b$ 。

(b) 证明对于每个素数 $p$ 都存在无限多个正高度可整除整数 $r$，使得 $pr$ 也是高度可整除的。