For-each-positive-integer-n-define-a-n-20-n-2-and-d-n-gcd-a-n-a-n-1-Find-the-set-of-all-values-that-are-taken-by-d-n-and-show-by-examples-that-each-of-these-values-are-attained-

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Question Number 22379 by Tinkutara last updated on 16/Oct/17

$$\mathrm{For}\mathrm{each}\mathrm{positive}\mathrm{integer}n,\mathrm{define}{a}_n=$$$$20+n^2,\mathrm{and}{d}_n=gcd(a_n,a_{n+1}).\mathrm{Find}$$$$\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{values}\mathrm{that}\mathrm{are}\mathrm{taken}\mathrm{by}$$$$d_n\mathrm{and}\mathrm{show}\mathrm{by}\mathrm{examples}\mathrm{that}\mathrm{each}\mathrm{of}$$$$\mathrm{these}\mathrm{values}\mathrm{are}\mathrm{attained}.$$

Answered by Tinkutara last updated on 21/Oct/17

$$Since{d}_n=gcd(a_n,a_{n+1})$$$$So{d}_{n}divides20+n^2.$$$$Similarly{d}_{n}divides20+{(n+1)}^2.$$$$So{d}_{n}divides{(n+1)}^2-n^2=2n+1$$$$\Rightarrow d_{n}divides4(20+n^2)=$$$$(2n+1)(2n-1)+81$$$$Since{d}_{n}alreadydivides2n+1,d_n$$$$mustdivide81.$$$$Hence{d}_{n}issetofalldivisorsof81.$$$$\therefore Requiredset=\{1,3,9,27,81\}$$

Commented by Rasheed.Sindhi last updated on 22/Oct/17

$$\mathrm{Nice}!$$

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