Prime Numbers : GMAT Problem Solving (PS)
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# Prime Numbers

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Manager
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13 Aug 2011, 17:39
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How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.
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13 Aug 2011, 17:56
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maheshsrini wrote:
How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.

24! = (1x2x3x4x5x6x7x...........24), which is divisible by 2, 3, 4, 5, 6, 7 ......24, and many other integer/number

- 24!+2 is not a prime because it is at least divisible by 2.
- 24!+3 is also not a prime because it is at least divisible by 3.
- 24!+4 is also not a prime because it is at least divisible by 2 and 4.
- 24!+5 is also not a prime because it is at least divisible by 5.
- 24!+6 is also not a prime because it is at least divisible by 2 and 3.

So there is zero prime between 24!+2 and 24!+6.
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13 Aug 2011, 23:06
maheshsrini wrote:
How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.

All prime numbers greater than 5 have to follow (6n+1) format, where n = 1,2, 3.. example - 5,7, 11, 13 etc.

Regarding above question - we've to worry only about - (24!+6) - 1
Since,(24!+6) - 1 = (24!+5) - which is divisible by 5. Hence, no prime number.

Hope it helps.

Cheers,
Aj.
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14 Aug 2011, 08:43
Ajay369 wrote:
maheshsrini wrote:
How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.

All prime numbers greater than 5 have to follow (6n+1) format, where n = 1,2, 3.. example - 5,7, 11, 13 etc.

This statement is not true. For example, 17, which does not fit to the format (6n+1).
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14 Aug 2011, 09:06
there is a small addition which our frnd AJ missed out... its generally 6n+1 or 6n-1
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14 Aug 2011, 11:11
viks4gmat wrote:
there is a small addition which our frnd AJ missed out... its generally 6n+1 or 6n-1

Thanks Vikas, i was about to respond.
Aforesaid, Prime number follows generally 6n+1 or 6n-1 format.
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14 Aug 2011, 17:47
Ajay369 wrote:
viks4gmat wrote:
there is a small addition which our frnd AJ missed out... its generally 6n+1 or 6n-1

Thanks Vikas, i was about to respond.
Aforesaid, Prime number follows generally 6n+1 or 6n-1 format.

Ok..... Lets be careful in saying that prime numbers over 5 are in these format: (6n+1) or (6n-1) but these format are not the ways to find prime numbers.

Because, these expressions are not true for every n in (6n+1) or (6n-1). For example if n = 6, then 6n-1=35, which is not a prime. Similarly when n=8, 6n+1=48. which is also not a prime.
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15 Aug 2011, 05:28
maheshsrini wrote:
How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.

I don't remember it exactly:

There can be NO prime number between n!+2 and n!+n, where n>2. Correct me if I am wrong.
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18 Aug 2011, 17:02
fluke wrote:
maheshsrini wrote:
How do you solve this problem algebrically?.

How many prime numbers are there between 24!+2 and 24!+6.

I don't remember it exactly:

There can be NO prime number between n!+2 and n!+n, where n>2. Correct me if I am wrong.

You are correct. (n!+p), where p is a prime>n>2, is a prime number.

For example, if n = 5 and p = 7, (n!+p) = 127, which is a prime...

Probably (n!+1) is also a prime where n>4.
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18 Aug 2011, 19:28
i agree with fluke.

there will be no prime numbers between n!+2 and n!+n where n>2.

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19 Aug 2011, 11:06
Fistail wrote:
Probably (n!+1) is also a prime where n>4.

Fistail for the part (n! +1 ) is also a prime where n>4, if n =5, 5! +1 = 120+1 = 121 which is 11^2 and this is not a prime..
7! +1 = 5040 +1 = 5041 = 71 ^2

so either this cant be a rule or looks like there are exceptions to this rule and i hit the exceptions part bang on its head twice :D
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19 Aug 2011, 12:35
viks4gmat wrote:
Fistail wrote:
Probably (n!+1) is also a prime where n>4.

Fistail for the part (n! +1 ) is also a prime where n>4, if n =5, 5! +1 = 120+1 = 121 which is 11^2 and this is not a prime..
7! +1 = 5040 +1 = 5041 = 71 ^2

so either this cant be a rule or looks like there are exceptions to this rule and i hit the exceptions part bang on its head twice :D

I had different thinking in my mind and was not very much sure. Therefore I said "probably"...
I take back my statement...

Thanks...
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19 Aug 2011, 22:40
Fistail wrote:
viks4gmat wrote:
Fistail wrote:
Probably (n!+1) is also a prime where n>4.

Fistail for the part (n! +1 ) is also a prime where n>4, if n =5, 5! +1 = 120+1 = 121 which is 11^2 and this is not a prime..
7! +1 = 5040 +1 = 5041 = 71 ^2

so either this cant be a rule or looks like there are exceptions to this rule and i hit the exceptions part bang on its head twice :D

I had different thinking in my mind and was not very much sure. Therefore I said "probably"...
I take back my statement...

Thanks...

naah dont worry... its always gud to think up new logics and post it out... there are a lot of ppl there who are looking for such shortcuts to nail the quant.. so no pressure at all.

just want to know your thought process behind this conclusion.. this may come in handy sometime or the other for all of us... - after all, for all we know 5! and 7! could be the only exceptions to ur rule!! (unless someone actually goes about finding factorials for 40321 and so on - fyi these are not divisible by 2-9, 11 or 19 . I lost my patience and dint go any further
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20 Aug 2011, 15:03
viks4gmat wrote:
naah dont worry... its always gud to think up new logics and post it out... there are a lot of ppl there who are looking for such shortcuts to nail the quant.. so no pressure at all.

just want to know your thought process behind this conclusion.. this may come in handy sometime or the other for all of us... - after all, for all we know 5! and 7! could be the only exceptions to ur rule!! (unless someone actually goes about finding factorials for 40321 and so on - fyi these are not divisible by 2-9, 11 or 19 . I lost my patience and dint go any further

In fact, it was my hasty conclusion, without having second thought. I knew I was not sure but had a quick thought that 5! is divisible by 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120 (I might have missed some numbers), 5!+1 is not divisible by none of these except 1. What I forgot is that there could numbers other than mentioned above. That what I missed..

Lesson learned - Always check twice before confirming answer......
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01 Sep 2011, 21:47
No prime Numbers between these two factorial because these two are multiples of 2 and 6 respectively.
Re: Prime Numbers   [#permalink] 01 Sep 2011, 21:47
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