MathDB

Problem 3

Part of 2019 LIMIT Category B

Problems(2)

factors of positive integer, find sum of reciprocals

Source: LIMIT 2019 CAS2 P6

4/28/2021
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}
number theory
ringness of sets

Source: LIMIT 2019 CBS2 P3

4/28/2021
A subset WW of the set of real numbers is called a ring if it contains 11 and if for all a,b∈Wa,b\in W, the numbers a−ba-b and abab are also in WW. Let S={m2n∣m,n∈Z}S=\left\{\frac m{2^n}|m,n\in\mathbb Z\right\} and T={pq∣p,q∈Z,q odd}T=\left\{\frac pq|p,q\in\mathbb Z,q\text{ odd}\right\}. Then <spanclass=′latex−bold′>(A)</span> neither S nor T is a ring<span class='latex-bold'>(A)</span>~\text{neither }S\text{ nor }T\text{ is a ring} <spanclass=′latex−bold′>(B)</span> S is a ring, T is not a ring<span class='latex-bold'>(B)</span>~S\text{ is a ring, }T\text{ is not a ring} <spanclass=′latex−bold′>(C)</span> T is a ring, S is not a ring<span class='latex-bold'>(C)</span>~T\text{ is a ring, }S\text{ is not a ring} <spanclass=′latex−bold′>(D)</span> both S and T are rings<span class='latex-bold'>(D)</span>~\text{both }S\text{ and }T\text{ are rings}
abstract algebra