IMO SL 2005 N5

For discussing Olympiad Level Number Theory problems
rah4927
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IMO SL 2005 N5

Unread post by rah4927 » Fri Feb 03, 2017 8:21 pm

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.

Source : IMO SL 2005 N5

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