OG question

jc114 wrote:153. Does the integerK have a factor p such that 1<p<k?

(1) k>4!

(2) 13! + 2<= K<=13!+13

Is it B ?

1 - insufficient. Take 29, it doesn't have a factor p that satisfies 1<p<k.
Take 30 - it does have a factor p that satisfies 1<p<k

2 - sufficient. 13!+2 is divisible by 2, 13!+3 is divisible by 3 etc. All values
in the range of 13!+2 to 13!+13 have atleast 1 factor that is > 1 and
less than k.

jc114 wrote: 154. Is x negative?

(1) x^3(1-x^2)<0

(2) x^2-1<0

Are there ways to solve without having to plug in #s?

I think plugging in values will work faster in this case. Else, it
will be easier to get into an algebraic mess

1 - insufficient. Take 2 and -0.5 for example

2 - insufficient. x^2 < 1. x should be in the range -1 to +1.
Take 0.5 and -0.5 for example

1 and 2 together is sufficient. 2 tells us that x has a range of -1 to +1.
The only values of x in this range that would satisfy condition 1 is if
-1 < x < 0.

So, x is negative or so I think . Please confirm.

154 IMO A

1. X3( 1-X2) <0

X > 1

Suff


2. -1<x<1

insuff

The answer is C

Statement 1:
statement 1 is satisfied we consider x= 2 or x= -1/2
So X can be either a +ve integer or -ve fraction

NOT SUFF
Statement 2:
statement 2 is satisfied we consider x= +1/2 or x= -1/2
So X can be either a +ve fraction or -ve fraction

NOT SUFF

Combined together, we know from statement two that X is a fraction and not an integer, and statement 1 gives us the information that x is a -ve fraction

Answer - C