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prime numbers explanation

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prime numbers explanation [#permalink] New post 20 Jul 2012, 17:20
Can someone pls explain me the significance of : If is a positive integer greater than 1, then there is always a prime number with n<n<2n

also, all prime numbers above 3 are of the form 6n-1 or 6n+1 , because all other numbers are divisible by 2 or 3. examples to both will be of great help

regards
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Re: prime numbers explanation [#permalink] New post 20 Jul 2012, 17:36
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sdas wrote:
Can someone pls explain me the significance of : If is a positive integer greater than 1, then there is always a prime number with n<n<2n

also, all prime numbers above 3 are of the form 6n-1 or 6n+1 , because all other numbers are divisible by 2 or 3. examples to both will be of great help

regards


This is the Bertrand's postulate: For every n > 1 there is always at least one prime p such that n < p < 2n.
The proof is complicated. It has no significance as far as GMAT is concerned. You are not expected to know Bertrand's postulate and this is not an intuitive number property.

The second property you mentioned is important. This is how you can explain it.

Every integer equal to or greater than 6 will be of one of the 6 forms: 6n or 6n+1 or 6n+2 or 6n+3 or 6n+4 or 6n+5

Note here that 6n is divisible by 6.
6n+2 = 2(3n+1) i.e. divisible by 2
6n+3 = 3(2n+1) i.e. divisible by 3
6n+4 = 2(3n+2) i.e. divisible by 2

Hence none of these can be prime. So a prime must be of the form 6n+1 or 6n+5 (which is the same as 6n-1)

Notice that 5 can be written as 6n - 1 where n = 1.
Only 2 and 3 cannot be written in one of the two forms: 6n+1 or 6n - 1.

Hence every prime number greater than 3 will be of one of these forms. (mind you, it does NOT mean that every number of the form 6n+1 or 6n-1 will be prime)
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Senior Manager
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Re: prime numbers explanation [#permalink] New post 22 Jul 2012, 14:37
Awesome. that helps. Thank you Karishma
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-----------------------------
Quant Notes consolidated: consolodited-quant-guides-of-forum-most-helpful-in-preps-151067.html#p1217652

My GMAT journey begins: my-gmat-journey-begins-122251.html

All about Richard Ivey: all-about-richard-ivey-148594.html#p1190518

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Joined: 23 Mar 2011
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Re: prime numbers explanation [#permalink] New post 22 Jul 2012, 15:07
If a is a factor of bc, and gcd(a,b)=1 then a is a factor of c
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-----------------------------
Quant Notes consolidated: consolodited-quant-guides-of-forum-most-helpful-in-preps-151067.html#p1217652

My GMAT journey begins: my-gmat-journey-begins-122251.html

All about Richard Ivey: all-about-richard-ivey-148594.html#p1190518

Senior Manager
Senior Manager
User avatar
Joined: 23 Mar 2011
Posts: 474
Location: India
GPA: 2.5
WE: Operations (Hospitality and Tourism)
Followers: 11

Kudos [?]: 107 [0], given: 59

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Re: prime numbers explanation [#permalink] New post 23 Jul 2012, 15:22
OK. got this one. also, if gcd(a,b) is 1 then both numbers are relatively prime
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-----------------------------
Quant Notes consolidated: consolodited-quant-guides-of-forum-most-helpful-in-preps-151067.html#p1217652

My GMAT journey begins: my-gmat-journey-begins-122251.html

All about Richard Ivey: all-about-richard-ivey-148594.html#p1190518

Re: prime numbers explanation   [#permalink] 23 Jul 2012, 15:22
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