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

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Math Expert
Joined: 02 Sep 2009
Posts: 60647
Let n be a positive integer. Is n^2 + 1 divisible by n + 1?  [#permalink]

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02 Dec 2019, 00:40
00:00

Difficulty:

55% (hard)

Question Stats:

55% (02:07) correct 45% (02:09) wrong based on 33 sessions

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

(1) n > 13

(2) n is prime

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Math Expert
Joined: 02 Aug 2009
Posts: 8336
Re: Let n be a positive integer. Is n^2 + 1 divisible by n + 1?  [#permalink]

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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

D
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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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