Question:
If x is a positive integer, how many positive integers less than x are divisors of x ?
(1) x^2 is divisible by exactly 4 positive integers less than x^2.
(2) 2x is divisible by exactly 3 positive integers less than 2x.Hi !
I think some questions like this are elegant ways to test on the test takers concept and ability to comprehend the question clearly!
Such questions fall under hard or even super hard category
because of the gap in the way we think and that which is expected by the GMAT.Let's break this down-
First,
GMAT Track of thought 1Look at the question stem and deep think.
What are we being tested on ? "How many factors.." So the question is a testing me on the number of factors and here it shall be total factors-1 since the total factors include the number x itself and the stem wants you to compute total factors less than x.
So an underlying fundamental concept has to be known and applied here-
that the total factors of any number lie between 1 & the number itself.
So, what exactly do I need to find? How do I simplify the stem?
I have to find the total number of factors of x in order to answer the question stem that wants me to compute (total number of factors of x-1). Now, lets move to the statements-
GMAT Track of thought 2St(1)- \(x^2 \)is divisible by exactly 4 positive integers less than \(x^2\).If you have been able to break down the question stem, analysing the statement should not be much of a trouble since its exactly the same thought process that you are called to apply.
What does this statement implies? How many factors \(x^2 \)has?
The statement indicates that \(x^2\) has 5 factors.
How would I represent a number with 5 factors? You need to
know a concept here that
If a number x = \(a^p\) * \(b^q\) * \(c^r\)... then it has total factors (p+1)*(q+1)*(r+1) where a,b,c are prime factors.(Suggestion- Don't remember this concept. Go into the depths of understanding how the concept is developed. Use it to analyse number of even and odd factors too. Bottom line- Dont remember, understand and internalise
)
Coming back to where I left,
How would I represent a number with 5 factors?
I want you to reverse engineer here and think of "what would be the prime factored form of x^2 if it has 5 factors"?
5=5*1
So\( x^2\) has to be in the form of (some prime number)^4
And if\( x^2\) is in the format of (some prime number)^4 , what would be the prime factored format of x? It shall be (some prime number)^2
So how many factors will x have? 3 ! And its a definite answer since there is no other format. (As 5=5*1)
This answers my question stem!
Sufficient. Eliminate B,C,E. We are down to 2 choices. Its A or D.
GMAT Track of thought 3St(2)2x is divisible by exactly 3 positive integers less than 2x.What does this statement implies? How many factors does 2x have ?
The statement indicates that 2x has 4 factors.
How would I represent a number with 4 factors?Again,
reverse engineer and ask yourself,
given 4 factors for a number, how would I represent the number in its prime factored form?2x =\( (some prime number)^3 \)or 2x = \((prime number 1)^1 \)* \((prime number 2)^1\)
What you have to cautious here is that you have a
2 in the number 2x.So, one of the primes is 2 already.
If 2 * x = \((some prime number)^3\) then x is \(2^2\) so that LHS= RHS. Thus number of factors x has is (2+1) or 3 factors.
Or
2x = \((prime number 1)^1\) * \((prime number 2)^1 \).Of the two prime numbers in the RHS, 2 is already one of them.
So, x is the other prime.
This implies x = \((prime number )^1\). Thus number of factors x has is (1+1)=2 factors.
We now have two different answers.
Insufficient.
Eliminate D.
The
correct option is A.If you practice analysing,deep thinking before jumping to statements and have a solid grip on the concepts tested on GMAT, you will enjoy solving such questions.
Difficulty is inversely proportional to the skills you acquire!
Don't feel anxious on the tags of 700+ over a question. Questions are to assess your prep.
If it detects the gaps in your GMAT readiness, fix and move.
Hope my explanation makes sense.
Let me know if you have questions.
Looking forward to help you.
Devmitra Sen
Head of Academics,CrackverbalConnect with me on Linkedin here