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factors of positive integer, find sum of reciprocals

Source: LIMIT 2019 CAS2 P6

April 28, 2021
number theory

Problem Statement

Let d1,d2,…,dkd_1,d_2,\ldots,d_k be all factors of a positive integer nn including 11 and nn. If d1+d2+…+dk=72d_1+d_2+\ldots+d_k=72 then 1d1+1d2+…+1dk\frac1{d_1}+\frac1{d_2}+\ldots+\frac1{d_k} is <spanclass=′latex−bold′>(A)</span> k272<span class='latex-bold'>(A)</span>~\frac{k^2}{72} <spanclass=′latex−bold′>(B)</span> 72k<span class='latex-bold'>(B)</span>~\frac{72}k <spanclass=′latex−bold′>(C)</span> 72n<span class='latex-bold'>(C)</span>~\frac{72}n <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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