Integers

[danjuma]

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

a. K > 4!

b.13! + 2 <=K<=13! + 13

[Rezinka]

Lets see..
We have an integer 'k' and its factor p such that 1<p<k

Statement 1 :
K>4!
or, k>24
If k = 25, it has a factor 5 where 1<5<25
but if k = prime, it will not have any factor which is greater than 1 and less than k as a prime number has only 2 factors i.e. 1 and itself
INSUFFICIENT

Statement 2 :
(13!+2) <= k <= (13!+13)
Now,
if k = 13! +2;
or, k = 13*12*11*10*9*8*7*6*5*4*3*2*1 + 2
or k = 2 (13*12*11*10*9*8*7*6*5*4*3**1 + 1)
so, 2 is a factor and 1<2<k
Similarly for all other numbers from 13!+2 to 13!+13 we will have a factor
So, SUFFICIENT

ANS : B

[goyalsau]

Lets see..
We have an integer 'k' and its factor p such that 1<p<k

Statement 1 :
K>4!
or, k>24
If k = 25, it has a factor 5 where 1<5<25
but if k = prime, it will not have any factor which is greater than 1 and less than k as a prime number has only 2 factors i.e. 1 and itself
INSUFFICIENT

Statement 2 :
(13!+2) <= k <= (13!+13)
Now,
if k = 13! +2;
or, k = 13*12*11*10*9*8*7*6*5*4*3*2*1 + 2
or k = 2 (13*12*11*10*9*8*7*6*5*4*3**1 + 1)
so, 2 is a factor and 1<2<k
Similarly for all other numbers from 13!+2 to 13!+13 we will have a factor
So, SUFFICIENT

ANS : B

Good work Buddy,
I thought the answer is E,

We have to proof that p is a factor of k
and we are given that p is less than k and its +ve

If we take statement II and assume that the value of p is 14
and take all the value from +2 to +13, But still not able to divide it completely,

Please help me understand what i am doing wrong.

[Rezinka]

The thing is :
We need to find if there exists a factor p of k that is greater than 1 and less than k

While 14 may not be a factor of k from 13!+2 to 13!+13; there are other factors.
Where you are going wrong is in that the question is not asking us to find a particular factor of k but is asking if there is ANY factor. So, while 14 might not be the factor in any case, we have 2,3,4,5,,6,7,8,9,20,11,12,13 as factors.

You want further explanation? I could show step-by-step procedure.