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-

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} .

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

a_{n} = 20 + n^{2}, d_{n} = \gcd (a_{n}, a_{n + 2})

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∵ d_{n} = \gcd (a_{n}, a_{n + 2})

∴ d_{n} ∣ 20 + n^{2} \land d_{n} ∣ 20 + {(n + 2)}^{2}

∴ d_{n} ∣ {20 + {(n + 2)}^{2}} - (20 + n^{2})

d_{n} ∣ 4 (n + 1) \Rightarrow d_{n} ∣ (n + 1) \dots \dots . . (i)

Now, 20 + n^{2} = (n + 1) (n - 1) + 21

So d_{n} ∣ 20 + n^{2} \Rightarrow d_{n} ∣ (n + 1) (n - 1) + 21 \dots . (ii)

(i) & (ii) : d_{n} ∣ 21

∴ The required set is {1, 3, 7, 21}

(Method of proof used by Mr Tinkutara in

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