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number theory

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Joined: 30 Oct 2017
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number theory [#permalink] New post 30 May 2018, 00:44
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Question Stats:

50% (00:47) correct 50% (01:21) wrong based on 2 sessions
If n is a positive integer and (n + 1)(n + 3) is odd, then (n + 2)(n + 4) must be a multiple of which one of the following?

(A) 3 (B) 5 (C) 6 (D) 8 (E) 16
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Re: number theory [#permalink] New post 01 Jun 2018, 15:31
Expert's post
shahul wrote:
If n is a positive integer and (n + 1)(n + 3) is odd, then (n + 2)(n + 4) must be a multiple of which one of the following?

(A) 3 (B) 5 (C) 6 (D) 8 (E) 16



For \((n + 1)(n + 3)\) to be odd \((n+1)\) is odd and \((n+3)\) is also odd.

Since only \(odd \times odd=odd\). So \(n\) must be even since only \(even + odd = odd\).

\((n + 2)(n + 4)= 2 \times (\frac{n}{2}+1) \times 2 \times (\frac{n}{2}+2)\).

Either \((\frac{n}{2}+1)\) is even or \((\frac{n}{2}+2)\) is even.

So this means there must be a factor of 8 (\(2 \times 2 \times 2\)).
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Re: number theory   [#permalink] 01 Jun 2018, 15:31
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