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Prime Number

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Prime Number [#permalink] New post   22 Feb 2011, 04:22
all prime numbers above 3 are of the form 6n-1or 6n+1, because all other numbers are divisible by 2 or 3.

Please make clear the meaning of the sentence with example.



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Re: Prime Number [#permalink] New post   22 Feb 2011, 05:46
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Baten80 wrote:
all prime numbers above 3 are of the form 6n-1or 6n+1, because all other numbers are divisible by 2 or 3.

Please make clear the meaning of the sentence with example.


Any prime number \(p>3\) when divided by 6 can only give remainder of 1 or 5 (remainder can not be 2 or 4 as in this case \(p\) would be even and remainder can not be 3 as in this case \(p\) would be divisible by 3).

So any prime number \(p>3\) could be expressed as \(p=6n+1\) or \(p=6n+5\) (\(p=6n-1\)), where n is an integer >0.

For example: 5=prime=6-1, 7=prime=6+1, 11=prime=6*2-1, 13=prime=6*2+1, 17=prime=6*3-1, ...

But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are prime, so vise-versa of above property is not correct. For example 25 yields a remainder of 1 upon division be 6 and it's not a prime number.
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Re: Prime Number [#permalink] New post   28 Feb 2011, 20:51
Thanks Bunuel ! Amazing explanation.
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Re: Prime Number [#permalink] New post   28 Feb 2011, 21:20
Bunuel
Any OG question using this property?

Thanks

Bunuel wrote:
Baten80 wrote:
all prime numbers above 3 are of the form 6n-1or 6n+1, because all other numbers are divisible by 2 or 3.

Please make clear the meaning of the sentence with example.


Any prime number \(p>3\) when divided by 6 can only give remainder of 1 or 5 (remainder can not be 2 or 4 as in this case \(p\) would be even and remainder can not be 3 as in this case \(p\) would be divisible by 3).

So any prime number \(p>3\) could be expressed as \(p=6n+1\) or \(p=6n+5\) (\(p=6n-1\)), where n is an integer >0.

For example: 5=prime=6-1, 7=prime=6+1, 11=prime=6*2-1, 13=prime=6*2+1, 17=prime=6*3-1, ...

But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are prime, so vise-versa of above property is not correct. For example 25 yields a remainder of 1 upon division be 6 and it's not a prime number.




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Re: Prime Number [#permalink]
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