Does there exist an integer d such that x > d > 1 & x/d is an integer?

Composite number is a number that has more than 2 distinct factors.

Question REPHRASED : Is x a composite Number ?

Statement 1: 11! + 2 < x < 11!+ 12

i.e. x = 11! + 2 but 11! is a multiple of 2 i.e. x is a multiple of 2 hence Yes a Composite Number
i.e. x = 11! + 3 but 11! is a multiple of 3 i.e. x is a multiple of 3 hence Yes a Composite Number
.....
..... and so on....
i.e. x = 11! + 12 but 11! is a multiple of 12 i.e. x is a multiple of 12 hence Yes a Composite Number

i.e. x is always a composite number. Hence,
SUFFICIENT


Statement 2: x > 2^5
x is greater than 32
i.e. x may or may not be a Composite Number
NOT SUFFICIENT

Answer: Option A

Hello

In statement 2, you are correct that many values exist for x>32 for which we will be able to find an integer 'd' such that x/d is an integer.
But lets look at the question. The question says, "Does there exist an integer d such that x > d > 1 and x/d is an integer?"

Unless we know what is 'x', how do we know whether such a 'd' exists or not. We are given that x>32. Say x=33, then definitely there exists a 'd' (3 or 11). But if x=41, then there does not exist such a 'd'. So it comes down to just one thing: we are NOT able to answer with a clear-cut YES or a clear-cut NO. Thats why statement 2 is not sufficient.

As for the highlighted part in your quote (I have highlighted), I think understanding questions and explanations here on Gmatclub will be the best thing to do. Ask questions, and try to understand the replies, maybe that will help you understand steadily over a short period of time.

You are correct for this assumption (it's more of a given) as it is given that d is an integer and x/d is also an integer. Thus these conditions are only possible when x is also an integer.

Hi,
it is not given that x/d is an integer, we have to find if x/d is an integer.

The two things mean a lot different..
and I do not think if we are finding whether x/d is an integer, we can infer that x is an integer.
this will be wrong interpretation. The Q is better with 'x as integer' stated clearly.

In a DS question, you can either prove the existence with absolute certainty or prove the absence with absolute certainty. Either way, you will have a sufficient statement. The way I have approached is the latter and hence I assumed that in order for x/d to be an integer with d as integer, x must be an integer ---> not prime (based on statement 1). Your approach and mine are thus different ways of approaching the same thing.

In a DS Q, the biggest traps are these only, where you are expected to believe/assume a variable as an integer.
the same Q with a choice
1. x is a reciprocal of a prime number.
here you can say that x/d is not an integer since the numerator is a fraction and the denominator is an integer...
This statement will be sufficient to tell you that x/d is not an integer and will be suff..

I was confused by the wording of this question. What does "Does there exist an integer" exactly mean? Does it mean, "Is there at least one integer that exists?", or does it mean "Is there one and only one integer that exists?"

It would mean atleast one and not one only. Had it been 'one only', the question would have stated such a thing.

I had a quick question in regards to the "does there exist an integer" portion of the question.

From this given info, I can gather that the question is asking if there is atleast one integer D that makes X/D an integer. I understand why statement 1 is SUF but could one make the argument that statement 2 could also be SUF making the answer choice D?

If X > 2^5, there is at least one value for X, despite the fact that X can be a composite or prime, where X/D is an integer. I understand that this question is testing the properties of prime numbers but the "does there exist an integer" portion is throwing me a loop.

Hi

The second statement is NOT sufficient. Ok x > 2^5 or x > 32.
Now if x = 33, then of course there exist a few integers between 1 and x which are factors of x (3, 11)
But if x = 37, then since x is prime, there is NO integer between 1 and x which is a factor of x.

we are not given whether x is prime or not. So second statement does not suffice.

Hi.

But when we say x may or may not have a prime number but still there is an integer d ( i.e. for x =36, integer d is 9) such that x/d is an integer.

Please help.

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Hi

Your query is not clear? Can you please rephrase what you want to know?

from first statement, it is clear that X is a composite number. So definitely there would be a positive integer greater than 1 but less than X which will be a factor of X.

amanvermagmat

Thanks for getting back.

I think I usually get too stuck with the language of question.

My doubt was again with the question stem, since question stem says " does there exist an integer d" from which I understand to be,if there is any one such value that makes statement ( either statement 1 or 2) true, then it is sufficient to answer that yes! The value exists.

In statement 2, like I mentioned , there are many values that exist for x>32 and for which x/d is an integer. Isnt it sufficient to answer that "yes! Value exists for x>32 where d/x is integer. Hence statement is sufficient".

I understand the solution that is presented.
I will be thankful if I can be provided some useful link to clear my thoughts on language of quant questions posed.

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Thanks amanvermagmat for such a wonderful piece of advice and explanation.
+1 for the same

Hi chetan2u,

Can you please help me in this.

The question is basically asking if there is any number d such that x > d > 1 and x/d = integer. Obviously we can say that yes it does exist for Statement II. The question is not asking for all value of d. Also, if we are to take the logic that d cannot be fixed for Statement II, how can be we fix d for Statement I.

Hi amanvermagmat,

As per your logic, as we cannot decide on the value of x, you negated Statement II. However, for Statement I also, we can have x in Decimals or fractions. X cannot be fixed for Statement I also. So, as per your logic of Statement II we should negate option I also.

Please let me know if my understanding is wrong.

Hello Rahul

Yes your perspective is good. I took x to be an integer. Actually I followed the statement in Chetan's solution, where he says that 'x should be an integer logically'.

So yes I solved the entire question assuming 'x' to be an integer