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Prime Numbers

by kushboo1987 » Tue Jul 27, 2010 3:07 am
if z is an integer, is z prime?

I) 15! < z
II) 17! + 2 <= z <= 17! + 17

A) only I is sufficient to answer the question.

B)only II is sufficient to answer the question.

C) both I and II are needed to answer the question.

D)Each statement alone is sufficient to answer the question.

E) statements I and II together are not sufficient to answer the question.
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Source: — Data Sufficiency |
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by sumanr84 » Tue Jul 27, 2010 3:52 am
Don't know how to solve..my guess would be E.

Someone pls share strategy to attack this one..
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by outreach » Tue Jul 27, 2010 4:08 am
probably E
1 ans 2 combined, z can have prime and non prime values
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by selango » Tue Jul 27, 2010 6:13 am
IMO E

stmt1,

z>15!

It contains both prime and non primes.

Insuff

stmt2,

17! + 2 <= z <= 17! + 17

1!=1

2!=4

3!=6

4!=4

5!=0

6!=0

The unit digit of 17! surely ll be zero.

17! + 2 <= z <= 17! + 17

Unit digit ll from 0 to 17.It contains both prime and non primes.

Combining 1 and 2 ll give same result

Hence E
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by kvcpk » Tue Jul 27, 2010 6:23 am
I) 15! < z
NO Info about Z.. INSUFF

II) 17! + 2 <= z <= 17! + 17
17! = 1*2*3*...17
17!+2 = 2(1+1*3*..........17)
Hence
17! +2 is div by 2 .. not prime
Similarly,
17! +3 is div by 3.. Not prime
17! +4 is div by 4..not prime
.......
....
17! +17 is div by 17.. not prime

Hence SUFFICIENT

pick B
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by selango » Tue Jul 27, 2010 7:18 am
kvcpk wrote:I) 15! < z
NO Info about Z.. INSUFF

II) 17! + 2 <= z <= 17! + 17
17! = 1*2*3*...17
17!+2 = 2(1+1*3*..........17)
Hence
17! +2 is div by 2 .. not prime
Similarly,
17! +3 is div by 3.. Not prime
17! +4 is div by 4..not prime
.......
....
17! +17 is div by 17.. not prime

Hence SUFFICIENT

pick B
Cool buddy:)
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by gmat1011 » Tue Jul 27, 2010 8:45 am
I don't think it is B

II) 17! + 2 <= z <= 17! + 17

Both 17! + 2 and 17! + 17 are non-prime

z is a number either equal to one of those 2 non-primes or somewhere in between --- so Z would be non-prime when equal to 17! + 2 or 17! + 17 but in may be prime in the gap between the two

I think it is E
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by selango » Tue Jul 27, 2010 8:58 am
gmat1011 wrote:I don't think it is B

II) 17! + 2 <= z <= 17! + 17

Both 17! + 2 and 17! + 17 are non-prime

z is a number either equal to one of those 2 non-primes or somewhere in between --- so Z would be non-prime when equal to 17! + 2 or 17! + 17 but in may be prime in the gap between the two

I think it is E
Its B only

17!+2>=Z<=17!+17

17!+2=2[1+17.16.15...3..1]

17!+3=3[1+17.16.15...2..1]

Z is not prime from 17!+2 to 17!+17 as each term is divided by 2,3,4... to 17.

The technique here is taking out the common factor and checking for divisibility
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by gmat1011 » Tue Jul 27, 2010 9:17 am
yes! you are right selango and kvcpk

thats how I arrived at the conclusion that 17! + 2 and 17! + 17 are both non-prime but then did not extend the logic to the gap between the two... you can factor out all the numbers in the gap and use the number from the factorial...

sheesh... my bad.
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by kvcpk » Tue Jul 27, 2010 10:07 am
selango wrote: Cool buddy:)
Thank you ;)
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by Brian@VeritasPrep » Tue Jul 27, 2010 2:46 pm
Hello, everyone:

Great thread - it looks like you're all on board with why the correct answer here is B, but if you don't mind I'd also like to point out why I had an impossible time not butting in here!

If you can think about any factorial, it's divisible by every number within it, pretty much by definition. Let's use 4! as an example:

4*3*2*1 is divisible by 4, 3, 2, and 1

Now, when you take that factorial and add 1, it can no longer be divisible by any of the prime factors (or, really, factors >1) that it previously was:

4*3*2*1 = 24, add 1 and it's 25, which is not divisible by 4, 3, or 2

The reason behind that is that:

Every second number (2, 4, 6, 8, 10...) is divisible by 2
Every third number (3, 6, 9, 12...) is divisible by 3
Every fourth number (4, 8, 12, 16) is divisible by 4

and so on...

If you take factorial and add 1, it's off of each of those cycles. You'd need to add 2 to keep it even, add 3 to keep it divisible by 3, etc. Only adding 1 just takes it off of each cycle that it was originally on.

So, for this problem, if you add any number between 2 and 17, you'll keep it on one of its previous factor cycles: add 3 and it's still divisible by 3; add 5 and it's still divisible by 5, etc.

If you had added 1, then there's a chance that it would be prime, but the number is so huge that you wouldn't really be able to calculate it as it might catch another, larger prime factor somehow. If they wanted to make it really difficult, though, they could have added 18 as a possible additive term (17! + 18). That would not be prime...17! is divisible by 2, so adding any even number to it would keep it even. You'd need to add 1 or a prime number greater than 17 in order to have the possibility of a prime number.

The more you think about problems like these, the better equipped you'll be to think quickly about them on test day. Thanks for bringing this one up and keeping the thread alive!
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by gmatmachoman » Tue Jul 27, 2010 8:43 pm
Guys, I saw this problem in GMAT Supplement MAth review...

Yeah its B only...good one!!
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by nicolezl » Wed Jul 28, 2010 5:31 am
Good explanation Brian! So just to make sure, the general rule is:

Any factorial plus any of its factors except 1 is not prime.

e.g. 20! + 1 MIGHT be prime and is not divisible by any number 2-20, (20! + 2) through (20! + 20) are NOT prime, and 20! + any prime number greater than 20 MIGHT be prime

Is that right?
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by Brian@VeritasPrep » Wed Jul 28, 2010 9:19 am
Hey Nicole

Almost exactly right! There should be two exceptions, 1! and 2!. 1! + 1 = 2, which is prime, and 2! + 1 = 3, which is prime. But this will work for anything larger than that. The main rule is that a factorial-plus-one is no longer divisible by any of its previous factors - and I'm sure it's written down in some math book somewhere as a hard-and-fast "rule", but I like that we just came up with it ourselves using math logic. The more you can embrace the fact that math rules (unlike, say, grammar rules or traffic laws) are logically certain, the easier it is to remember and apply them because you can always go back to the logic behind them.

Now, for the GMAT application, you'll probably see them test this in a few ways:

1) Data Sufficiency like this question, in which you'll be able to prove that a number is not prime (but probably not the other way, at least not with big numbers, since proving that a large number is prime is pretty labor-intensive)

2) Problem Solving in which they could ask you for the smallest prime factor of a factorial-plus-one (which will be greater than any of the factors of the factorial, based on that "rule" we just proved).

One thing I love to do when finding rules or properties like this is to anticipate how the GMAT might be able to test them. That not only prepares you for those types of questions that you may see, but also gets you thinking like the authors of the test which is just a good frame of mind to keep so that you're looking at rules and properties from different angles.
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by pharmxanthan » Thu Jul 29, 2010 9:15 am
Brian@VeritasPrep wrote:
2) Problem Solving in which they could ask you for the smallest prime factor of a factorial-plus-one (which will be greater than any of the factors of the factorial, based on that "rule" we just proved).
So, if we have to find the smallest prime factor of 10!+1, then the answer can not be 2,3,5, or 7? Can we find the smallest prime factor of 10!+1?
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