IMO Shortlist 2005 problem N5


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April 2, 2012
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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: Međunarodna matematička olimpijada, shortlist 2005