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# What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)

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Math Expert
Joined: 02 Sep 2009
Posts: 59725
What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)  [#permalink]

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20 Mar 2019, 23:02
00:00

Difficulty:

65% (hard)

Question Stats:

49% (01:49) correct 51% (01:51) wrong based on 35 sessions

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What is the largest integer that is a divisor of $$(n+1)(n+3)(n+5)(n+7)(n+9)$$ for all positive even integers n?

(A) 3
(B) 5
(C) 11
(D) 15
(E) 165

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Posts: 3158
Re: What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)  [#permalink]

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20 Mar 2019, 23:12

Solution

Given:
• The number n is a positive even integer

To find:
• The largest integer that is a divisor of (n + 1) (n + 3) (n + 5) (n + 7) (n + 9)

Approach and Working:
As n is an even positive integer, each of the given terms (n + 1), (n + 3), (n + 5), (n + 7) and (n + 9) are odd integers.
• Also, as the gap between the consecutive terms is always 2, the numbers are consecutive odd integers.

Now, in any 5 consecutive odd positive integers,
• One of them is always divisible by 3
• And one of them is always divisible by 5.
• Therefore, their product is always divisible by 3 x 5 = 15

Hence, the correct answer is option D.

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Re: What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)  [#permalink]

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21 Mar 2019, 00:25
Bunuel wrote:
What is the largest integer that is a divisor of $$(n+1)(n+3)(n+5)(n+7)(n+9)$$ for all positive even integers n?

(A) 3
(B) 5
(C) 11
(D) 15
(E) 165

we can check by substituting values of n =2,4,6 so on
we observe that the largest integer which comes is 15
IMO D
Re: What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)   [#permalink] 21 Mar 2019, 00:25
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