Problem
NT-B2-M10-P011 Exactly 14 Divisors
#11
★★★★☆ Level 4 of 5
Describe all natural numbers that have exactly \(14\) positive divisors.
1001 Problems in Classical Number Theory (method inspiration) · Problem 475
Factor \(14\) as a product of numbers \(a_i+1\).
If \(n=\prod p_i^{a_i}\), then \(\tau(n)=\prod(a_i+1)=14\). Since \(14=14\) or \(14=7\cdot 2\), there are two types. First, \(n=p^{13}\). Second, \(n=p^6q\), where \(p\ne q\) are primes. There are no other types.
A short but important classification.