Does the integer k have a factor p such that 1<p<k?


(1) k > 4!

(2) \(13! + 2 \leq k \leq 13!+13\)

Thanks Bunuel. I have a question, what would happen if the second statement said this?

(2) \(13!+1\leq{k}\leq{13!+13}\)

I have seen some similar problems, in which they add 1. Could you post some links about it?

Thanks!
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Sorry for the question, but I am just not understanding how you get 8*(2*4*5*6*7*9*10*11*12*13+1) from 13!+8 then how do you know that k is a multiple of 8...I obviously have some deficiencies when it comes to number properties. So then how is k never a prime between those parameters?
I would think (1*2*3*4*5*6*7*8*9*10*11*12*13) +8 ...

Hi bunuel
I don't understand the definition of a prime ( 1<p<k ). I know what a prime is but this drives me nuts.
For an example I take 10 for k. I have factors 2x5 that are greater than 1 and smaller than 10.

Or is it asking for one factor? I see "a factor" in the question, which is singular. Then I understand this. It is really tricky.

Posted from GMAT ToolKit

Hi Bunuel, I have a small query here- Had the question been : 4!+2<k<4!+6, if I follow the approach explained above I may conclude that k is not a prime no. reason being all the factors of 4! are the factors of 4!+?. Since its easy to calculate here the values between 4!+2 and 4!+6, I already know that it includes 29 which is a prime no., where as in case of 13!+2<k<13!+13, it is not easy to calculate 13!, we may be missing some value which is a prime no. How do we make sure that we are not missing anything? (Assuming we can not cross check by calculating the values because of the time constraint at the exam time)

Essentially this is about prime and non-prime.
Option 1: k>4!
Now assume k=5. then is there any factor of 5 which lies between 1 and 5? NO.
Take k=6. then there is 2,3 which lies between 1 and 6.
So 1 alone insufficient.

Option 2: 13!+2<=k<=13!+13
Take say 13!+3. now it can be rewritten as 3*{(13.12.11.....4.2.1)+1}
So there is 3 as factor between 1 and k. Similarly you can take the common factor out of all nos. in option 2.
Therefore option 2 is sufficient!
Add Kudos if this helps!

Target question: Does the integer k have a factor p such that 1 < p < k ?

This question is a great candidate for rephrasing the target question. (We have a free video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100)

Let's look at a few cases to get a better idea of what the target question is asking.
- Try k = 6. Since 2 is a factor of 6, we can see that k DOES have a factor p such that 1<p<k.
- Try k = 10 Since 5 is a factor of 10, we can see that k DOES have a factor p such that 1<p<k.
- Try k = 16. Since 4 is a factor of 14, we can see that k DOES have a factor p such that 1<p<k.
- Try k = 5. Since 1 and 5 are the ONLY factors of 5, we can see that k does NOT have a factor p such that 1<p<k.
Aha, so if k is a prime number, then it CANNOT satisfy the condition of having a factor p such that 1 < p < k
In other words, the target question is really asking us whether k is a non-prime integer (aka a "composite integer")

REPHRASED target question: Is integer k a non-prime integer?

Statement 1: k > 4!
In other words, k > 24
This does not help us determine whether or not k is a non-prime integer? No.
Consider these two conflicting cases:
Case a: k = 25, in which case k is a non-prime integer
Case b: k = 29, in which case k is a prime integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 13! + 2 ≤ k ≤ 13! + 13
Let's examine a few possible values for k.

k = 13! + 2
= (13)(12)(11)....(5)(4)(3)(2)(1) + 2
= 2[(13)(12)(11)....(5)(4)(3)(1) + 1]
Since k is a multiple of 2, k is a non-prime integer

k = 13! + 3
= (13)(12)(11)....(5)(4)(3)(2)(1) + 3
= 3[(13)(12)(11)....(5)(4)(2)(1) + 1]
Since k is a multiple of 3, k is a non-prime integer

k = 13! + 4
= (13)(12)(11)....(5)(4)(3)(2)(1) + 4
= 4[(13)(12)(11)....(5)(3)(2)(1) + 1]
Since k is a multiple of 4, k is a non-prime integer

As you can see, this pattern can be repeated all the way up to k = 13! + 13. In EVERY case, k is a non-prime integer

Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
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"Question basically asks whether k is a prime number. If it is, then it won't have a factor psuch that 1<p<k

(definition of a prime number)."

Hi,

I just don't understand this. Can you please explain?

[quote="Bunuel"][quote="mneeti"][quote="Bunuel"]


Bunuel , Please make me understand one thing.

The question tells us that 1<p<k

which means p can be anything smaller than k , and question asks if p is a factor of K?

it means if k is 25 then is p a factor of 25? or we can assume p be any number less than 25 and check if its factor of 25 or not

My question is in option B,

If 13! + 2 <= k <= 13! + 13 , it means k is a very large digit and lies between those equalities.
SO NOW , p must be any integer less than k and we need to see if p is a factor of k?

How about 17? lets take p as 17

1<=17<=(13!+2,13!+3,13!+4........13!+13)
as we know from B that k has a factor , yes we can see 2, 3 ,4 ,5 till 13 it has factor but hat about 17?

17 satisfies the criteria and it should also be a factor and if its not then it must be uncertain if p is factor of k or not?


PLEASE HELP

Bunuel - i have subtracted 13! from each side of the inequality and got that k is equal or between 0 to 11. From this inequality i can't got the answer. Where am i wrong? By the way - i looked at the math book for inequality theory but haven't find something relevant. Is there any post that covers this topic?

Thanks in advance!