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If n is a positive integer, what is the maximum possible num [#permalink]
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09 Feb 2011, 14:14
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If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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09 Feb 2011, 17:01
for n=1, the numbers are 2,3,4,5,6,7
out if these 2,3,5,7 these 4 are prime numbers.
now for any value of n >1 all six consecutive numbers will be 3 odd and 3 even. so at max we will get 3 prime numbers(assuming all the odd numbers are prime)
so the ans is 4



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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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09 Feb 2011, 18:28
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loveparis wrote: 33. If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Every prime number greater than 3 is either of the form 6n  1 or of the form 6n + 1. This means that in any 6 consecutive numbers, there can be at most 2 prime numbers with a difference of 2 between them e.g. 11 (form 6n  1) and 13 (form 6n + 1). Only 2 and 3 are prime numbers that are not of one of these forms. So if you take 2, 3, 4, 5, 6, 7 as the 6 consecutive numbers, you get 4 of them prime, the maximum possible.
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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10 Feb 2011, 04:24
Hi Karishma How does this follow ? "This means that in any 6 consecutive numbers, there can be at most 2 prime numbers with a difference of 2 between them" Regards, Subhash
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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10 Feb 2011, 04:44
subhashghosh wrote: Hi Karishma
How does this follow ?
"This means that in any 6 consecutive numbers, there can be at most 2 prime numbers with a difference of 2 between them"
Regards, Subhash Karishma has rightly said that primes are of the form 6n+1 or 6n1 This is not a formula for primes, it's more of a 'check' If the number can be written in the form 6n+1 or 6n1, it's PROBABLY a prime e.g. Put n=4 6x4+1=25<<< Not Prime However, 6n1=23 <<< Prime! :D See? Now assume that for a particular value of n, 6n1 and 6n+1 yield primes, in such a case the difference between the two would be: 6n+1  6n+1 = 2 Thus, the minimum difference between two prime numbers is 2. Now consider 6 consecutive numbers: n, n+1, n+2, n+3, n+4, n+5 Assume the middle term is a multiple of 6. So, n+3 and n+2 might be primes. If n+2 and n+3 are primes, then n+1 and n+5 wouldn't be primes. Thus, the maximum number of primes (greater than 3) that can occur in 6 consecutive numbers can never be more than 2! I hope the explanation is clear :D
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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10 Feb 2011, 05:35
subhashghosh wrote: Hi Karishma
How does this follow ?
"This means that in any 6 consecutive numbers, there can be at most 2 prime numbers with a difference of 2 between them"
Regards, Subhash In any 6 consecutive numbers, you can have at most 2 numbers of the form 6n  1 or 6n + 1 e.g. say the 6 consecutive numbers are: 6n  1, 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4 (e.g. 5, 6, 7, 8, 9, 10) or 6n  2, 6n  1, 6n, 6n + 1, 6n + 2, 6n + 3 (e.g. 10, 11, 12, 13, 14, 15 ) etc Remember, every number of the form 6n  1 or 6n + 1 is not prime. e.g. 25 is of the form 6n + 1 but it is not prime. But every prime greater than 3 is of the form 6n  1 or 6n + 1.
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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loveparis wrote: 33. If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Please tag number properties
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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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27 Feb 2017, 04:03
In order to maximize the number of prime numbers in consecutive series, we have to include 2 and 3. To do this, we have to put n=0 and n=1. For n=0, the series is 1,2,3,4,5,6. Number of prime numbers are 3 For n=1, the series is 2,3,4,5,6,7. Number of prime numbers are 4 So maximum will be 4. Option D



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Re: If n is a positive integer, what is the maximum possible num [#permalink]
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01 Mar 2017, 17:11
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loveparis wrote: If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6 In solving this problem, we must recall that 2 is the only even prime number. Thus, when n = 1, we have: 2, 3, 4, 5, 6, and 7, which gives us 4 prime numbers (2, 3, 5, and 7). Since when n is greater than 1 we will have 3 odd numbers and 3 even numbers (all greater than 2), the maximum number of prime numbers we could have is 3. Thus, by letting n = 1, we have 4 prime numbers, which is the maximum number of primes we could have. Answer: C
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