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Let n be a positive integer. Is n^2 + 1 divisible by n + 1?

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Let n be a positive integer. Is n^2 + 1 divisible by n + 1?  [#permalink]

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New post 02 Dec 2019, 00:40
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Re: Let n be a positive integer. Is n^2 + 1 divisible by n + 1?  [#permalink]

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New post 02 Dec 2019, 04:22
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Let n be a positive integer. Is n^2 + 1 divisible by n + 1?

\(n^2+1=(n+1)^2-2n\). So we have to check whether 2n is divisible by n+1 or \(\frac{2n}{n+1}\) is an integer. This can only happen when n=1, as n and n+1 are co-prime.

Thus the question is whether \(n=1\).

(1) n > 13
No, so n^2+1 is NOT divisible by n+1.
Suff

(2) n is prime
No \(n\neq{1}\), so n^2+1 is NOT divisible by n+1.
Suff

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Re: Let n be a positive integer. Is n^2 + 1 divisible by n + 1?   [#permalink] 02 Dec 2019, 04:22
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