Prime number question.

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Incognito
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There exist at least one prime number between n and 9n/8. What is the minimum value of n???

anita1's picture
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9 has no factor of 2 so N

9 has no factor of 2 so N must be divisible by 8
Let it be 8a where a ≥ 1
Now as per the question there shd be one prime number between 8a and 9a
For a =1, there is no number
a=2 , we have 17 a prime number
So the answer is 16.

nishit's picture
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To cont your assumptionFor

To cont your assumption
For a=3 the number are 24, 25, 26 & 27 and none of them are prime.
So its wrong.

I am not going to prove it but this is one of the great works of Joseph Bertrand and according to him
if n is an integer greater than 3, then there is at least one prime between n and 2n-2

Few other important points are

  • If n is at least 48, then there is at least one prime between n and 9n/8.
  • If n is greater than 6, then there is at least one prime of theform 4k+1 and at least one of the form 4k+3 between n and 2n.
  • If n is greater than or equal to 118, then the interval n to 4n/3 inclusive contains a prime of each of the forms 4n+1, 4n-1, 6n+1, and 6n-1.
  • If n is greater than 15, then there is at least one number between n and 2n that is the product of three different primes.

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mahadevan's picture
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y cant d answer be 10???

y cant d answer be 10???

n and 9n/8 means between 10 and 11.25, right...
11 is a prime number....
so 10 can b d answer, right???

Incognito (not verified)
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Simply because 11.25 is not

Simply because 11.25 is not an integer

mahadevan's picture
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does the question specify

does the question specify dat it shud be an integer?

ashish chauhan (not verified)
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Prime number question

The answer is very simple.It is clear that n/8 should be an integer.Taking n/8=1,I get no prime number between 8 & 9.Then assuming n/8=2,I get a prime number 17 between 16 & 18,i.e.,between n & 9n/8.So, the correct answer is "n=2".

ashish chauhan (not verified)
Incognito's picture
Prime number question

The answer is very simple.It is clear that n/8 should be an integer.Taking n/8=1,I get no prime number between 8 & 9.Then assuming n/8=2,I get a prime number 17 between 16 & 18,i.e.,between n & 9n/8.So, the correct answer is "n=2".

Incognito
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The Answer Should

The answer should be guessed first that it is an integer or not,so what are the options anyway???

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