Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 22635 by Rasheed.Sindhi last updated on 21/Oct/17

For each positive integer n, define a_n =20+n^2 ,and d_n =gcd(a_n ,a_(n+2) ). Find the set of all values that are taken by d_n .

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n},\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{20}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} . \\ $$$$ \\ $$

Answered by Rasheed.Sindhi last updated on 21/Oct/17

a_n =20+n^2 , d_n =gcd(a_n ,a_(n+2) ) −−−−−−−−−−−−− ∵ d_n =gcd(a_n ,a_(n+2) ) ∴ d_n ∣ 20+n^2 ∧ d_n ∣ 20+(n+2)^2 ∴ d_n ∣ {20+(n+2)^2 }−(20+n^2 ) d_n ∣ 4(n+1)⇒d_n ∣ (n+1)........(i) Now, 20+n^2 =(n+1)(n−1)+21 So d_n ∣ 20+n^2 ⇒d_n ∣ (n+1)(n−1)+21....(ii) (i)&(ii): d_n ∣ 21 ∴The required set is{1,3,7,21} (Method of proof used by Mr Tinkutara in answer of Q# 22379)

$$\mathrm{a}_{\mathrm{n}} =\mathrm{20}+\mathrm{n}^{\mathrm{2}} ,\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right) \\ $$$$−−−−−−−−−−−−− \\ $$$$\because\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right) \\ $$$$\therefore\:\mathrm{d}_{\mathrm{n}} \:\mid\:\mathrm{20}+\mathrm{n}^{\mathrm{2}} \:\wedge\:\mathrm{d}_{\mathrm{n}} \:\mid\:\mathrm{20}+\left(\mathrm{n}+\mathrm{2}\right)^{\mathrm{2}} \\ $$$$\therefore\:\mathrm{d}_{\mathrm{n}} \:\mid\:\left\{\mathrm{20}+\left(\mathrm{n}+\mathrm{2}\right)^{\mathrm{2}} \right\}−\left(\mathrm{20}+\mathrm{n}^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\:\mathrm{d}_{\mathrm{n}} \:\mid\:\mathrm{4}\left(\mathrm{n}+\mathrm{1}\right)\Rightarrow\mathrm{d}_{\mathrm{n}} \mid\:\left(\mathrm{n}+\mathrm{1}\right)........\left(\mathrm{i}\right) \\ $$$$\mathrm{Now},\:\mathrm{20}+\mathrm{n}^{\mathrm{2}} =\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}−\mathrm{1}\right)+\mathrm{21} \\ $$$$\mathrm{So}\:\mathrm{d}_{\mathrm{n}} \:\mid\:\mathrm{20}+\mathrm{n}^{\mathrm{2}} \Rightarrow\mathrm{d}_{\mathrm{n}} \mid\:\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}−\mathrm{1}\right)+\mathrm{21}....\left(\mathrm{ii}\right) \\ $$$$\:\left(\mathrm{i}\right)\&\left(\mathrm{ii}\right):\:\mathrm{d}_{\mathrm{n}} \mid\:\mathrm{21}\:\:\:\:\:\: \\ $$$$\therefore\mathrm{The}\:\mathrm{required}\:\mathrm{set}\:\mathrm{is}\left\{\mathrm{1},\mathrm{3},\mathrm{7},\mathrm{21}\right\} \\ $$$$\left(\mathrm{Method}\:\mathrm{of}\:\mathrm{proof}\:\mathrm{used}\:\mathrm{by}\:\mathrm{Mr}\:\mathrm{Tinkutara}\:\mathrm{in}\right. \\ $$$$\left.\mathrm{answer}\:\mathrm{of}\:\mathrm{Q}#\:\mathrm{22379}\right) \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com