MathDB

5

Part of 2022 Cono Sur

Problems(1)

Differences of adjacent divisors are also divisors

Source: Cono Sur 2022 #5

8/9/2022
An integer n>1n>1 , whose positive divisors are 1=d1<d2<⋯<dk=n1=d_1<d_2< \cdots <d_k=n, is called <spanclass=′latex−italic′>southern</span><span class='latex-italic'>southern</span> if all the numbers d2−d1,d3−d2,⋯ ,dk−dk−1d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1} are divisors of nn.
a) Find a positive integer that is <spanclass=′latex−italic′>notsouthern</span><span class='latex-italic'>not southern</span> and has exactly 20222022 positive divisors that are <spanclass=′latex−italic′>southern</span><span class='latex-italic'>southern</span>.
b) Show that there are infinitely many positive integers that are <spanclass=′latex−italic′>notsouthern</span><span class='latex-italic'>not southern</span> and have exactly 20222022 positive divisors that are <spanclass=′latex−italic′>southern</span><span class='latex-italic'>southern</span>.
number theory