If x is a positive integer, how many positive integers less than x are

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.
\(x^2\) will be divisible by exactly 5 integers including \(x^2\), only when x is a perfect square of a prime number..
for example \(x=a^2..x^2=a^4\)... Number of factors = \((4+1)=5\)
So, x or \(a^2\) will have (2+1) or 3 factors, and our answer is 2 as we have to exclude x itself.
Suff

(2) 2x is divisible by exactly 3 positive integers less than 2x.
This can have two cases for 4 factors including 2x itself..
a) x is any other prime number other than 2...\(2^1x^1\) will have factors as (1+1)(1+1)=2*2=4.. x will have (1+1) or 2 factors and just 1 factor less than x itself.
b) x is also in terms of 2 or \(x=2^2\), so \(2x=2^3\)...factors =(3+1) or 4, and x or \(2^2\) will have (2+1) or 3 factors, and our answer is 2 as we have to exclude x itself.
Insuff


A
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Why do you assume that x=p^a*q^b ?
It is possible that x = p^a or x = p^a*q^b*r^c...
etc

You just need the number of factors of x and you can subtract 1 out of it to get the number of factors less than x.

(1) x^2 is divisible by exactly 4 positive integers less than x^2.

So x^2 has 5 factors. 5 cannot be a product of two numbers greater than 1 since it is a prime number.
So x^2 = p^4
to give x = p^2
So x will have total 3 factors and 2 factors less than x.
Sufficient

(2) 2x is divisible by exactly 3 positive integers less than 2x.

So 2x has 4 total factors.
4 = 4*1 = 2*2
2x = p*q (in this case, x is not 2 but some other prime number. So it will have exactly 1 factor smaller than itself)
or
2x = p^3 (in this case x = 2^2 so it has 2 factors smaller than itself)
We don't know whether x has 1 or 2 factors less than itself.
Not sufficient

Answer (A)
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Bunuel Could you please provide a simplified answer for this question??

Here is my solution. priya2810

Statement 1: x^2 = 16 has only 4 divisors, then x = 4 has only two divisors 1 and 2 excluding 4. Now, x^2 = 25, 36, or 49 all have less or more divisors than 4 so they do not work. Thus, the number of positive integers less than x that are divisors of x is 2.

Statement 2: take 2x=6 and 2x=8. The first x will equal 3 and second will equal 4. 3 has only 1 divisor less 3 and that is 1. But 4 has 1 and 2 as divisors less than 4. This makes this statement insufficient.

Answer is A.

Does that help?

Bunuel
chetan2u
VeritasKarishma
can u help me with this?

say x=p^a*q^b where p and q are prime numbers

now no of factors excluding x = (a+1)(b+1)-1= a+b+ab

1)x^2=p^2a*q^2b
now no of factors of x^2 excluding x^2
= (2a+1)*(2b+1)-1=4
implies a+b+2ab=2

so how to solve further
required is a+b+ab but i have a+b+2ab ?

Plz help

VeritasKarishma: Thanks for the explaination.

Very high quality question from OG Advanced!

Statement 1: x² has 5 positive factors
I.e. x = a² where a is prime
x has 3 factors and 2 if them are less than x
*Sufficient*

Statement 2 says that x may be 2² or an odd prime number.
So x may have 3 factors or 2 factors hence
*Not sufficient*

Answer: Option A

Solution:

If we can determine the number of divisors of x, then if we subtract 1 from this number, the result will be the number of divisors of x that are less than x.

Statement One Only:

x^2 is divisible by exactly 4 positive integers less than x^2.

This means x^2 has 5 divisors (including x^2 itself). In order to have 5 divisors, x^2 = p^4 for some prime number p. Therefore, x = p^2 and hence x has 2 + 1 = 3 divisors (including x itself). So there are 2 divisors of x that are less than x.

Statement one alone is sufficient.

Statement Two Only:

2x is divisible by exactly 3 positive integers less than 2x.

This means 2x has 4 divisors (including 2x itself). In order to have 4 divisors, 2x = p^3 for some prime number p or 2x = q * r for some distinct prime numbers q and r. In the former case, x = (p^3)/2. However, since x is a positive integer and p is a prime, p must be 2. Therefore, x = 2^3/2 = 4, and hence x has 3 divisors (namely, 1, 2 and 4). So there are 2 divisors of x that are less than x. In the latter case, x = (q * r)/2. Again, since x is a positive integer, either q or r is 2. If q = 2, then x = r and if r = 2, then x = q. Either way, x is a prime and has 2 divisors. So there is only 1 divisor of x that is less than x.

Statement two alone is not sufficient.

Answer: A
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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.

For x^2 to have exactly 5 factors, it needs to be a the square of a prime number.

For example \(x = 2^2; x^2 = 4^2 = 16\)

There are 2 factors of x that are less than x (1 and 2). SUFFICIENT.

(2) 2x is divisible by exactly 3 positive integers less than 2x.

This statement tells us that 2x has 4 divisors. x may be \(2^2\) or an odd prime number -- x may have 2 or 3 factors. INSUFFICIENT.

Answer is A.
_________________

Video solution from Quant Reasoning:
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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.

To find
Total factors of X minus 1

Concepts to remember
A square of a prime number has ODD NUMBER of total factors
X = P^a * Q^b * R^c... has total factors at (a+1)*(b+1)*(c+1) ... (P,Q,R are prime numbers after prime factorizing an integer)


Statement 1
X^2 has 4 positive integers less than itself - - - - > Total factors are 5 (including the no itself)!

Such 5 factors can be produced only if expression is in the form of a^4. So X^2 could be say 7^4. This is sufficient to calculate total no of factors minus 1 as the questions asks for. SUFFICIENT


Statement 2
2x has total factors of 3 less than itself - - - - > Total factors of 2x are 4 (including the no itself)!

4 factors can exist if expression is say X1 = P^3 or if X2 = P^1 * Q^1 ...(P,Q,R are prime numbers after prime factorizing an integer)

Thus X1/2 and X2/2 will have different total factors.

eg. 2 * X(say 3) = 6 - -> which has total 4 factors(namely 1,2,3,6). But, total factors of X (3 as per example) are 1.

eg. 2 * X(say 4) = 8 - -> which has total 4 factors (namely 1,2,4,8). But, total factors of X (2 as per example) are 2.

Hi BrentGMATPrepNow, Could your share your usual ways of easy to understand methods please? Thanks for your great helps always and time in advanced. Much appreciated

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 1

Look 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 2

St(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 3

St(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,Crackverbal
Connect with me on Linkedin here