Last visit was: 03 Aug 2024, 07:12 It is currently 03 Aug 2024, 07:12
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

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

SORT BY:
Tags:
Show Tags
Hide Tags
Senior Manager
Joined: 04 Sep 2017
Posts: 334
Own Kudos [?]: 20809 [213]
Given Kudos: 61
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11502
Own Kudos [?]: 34846 [52]
Given Kudos: 329
Intern
Joined: 27 Apr 2017
Posts: 14
Own Kudos [?]: 12 [11]
Given Kudos: 8
General Discussion
Current Student
Joined: 19 Jun 2019
Posts: 14
Own Kudos [?]: 18 [0]
Given Kudos: 16
Location: India
GMAT 1: 720 Q49 V40
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
Manager
Joined: 17 Jul 2017
Posts: 205
Own Kudos [?]: 97 [0]
Given Kudos: 228
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
gmatt1476 wrote:
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.

DS05541.01

Bunuel
chetan2u
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
Tutor
Joined: 16 Oct 2010
Posts: 15181
Own Kudos [?]: 67088 [10]
Given Kudos: 436
Location: Pune, India
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
10
Kudos
vanam52923 wrote:
gmatt1476 wrote:
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.

DS05541.01

Bunuel
chetan2u
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

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

PM Intern
Joined: 27 Feb 2019
Posts: 221
Own Kudos [?]: 183 [1]
Given Kudos: 197
Location: India
GMAT 1: 720 Q48 V41
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
1
Bookmarks
Manager
Joined: 20 Mar 2019
Posts: 66
Own Kudos [?]: 16 [0]
Given Kudos: 43
GMAT 1: 760 Q50 V42
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
Very high quality question from OG Advanced!
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6051
Own Kudos [?]: 13873 [3]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
3
Bookmarks
gmatt1476 wrote:
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.

DS05541.01

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*

Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19249
Own Kudos [?]: 22783 [9]
Given Kudos: 286
Location: United States (CA)
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
1
Kudos
8
Bookmarks
gmatt1476 wrote:
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.

DS05541.01

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.

Director
Joined: 09 Jan 2020
Posts: 953
Own Kudos [?]: 236 [0]
Given Kudos: 432
Location: United States
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
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.

Tutor
Joined: 17 Jul 2019
Posts: 1304
Own Kudos [?]: 1754 [9]
Given Kudos: 66
GMAT 1: 780 Q51 V45
GMAT 2: 780 Q50 V47
GMAT 3: 770 Q50 V45
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
7
Kudos
2
Bookmarks
Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
Manager
Joined: 25 Jan 2021
Posts: 242
Own Kudos [?]: 92 [1]
Given Kudos: 694
Location: India
GMAT 1: 760 Q49 V44
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
1
Kudos
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.
Senior Manager
Joined: 16 Nov 2021
Posts: 472
Own Kudos [?]: 28 [0]
Given Kudos: 5901
Location: United Kingdom
If x is a positive integer, how many positive integers less than x are [#permalink]
gmatt1476 wrote:
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.

DS05541.01

Originally posted by Kimberly77 on 03 Oct 2022, 02:45.
Last edited by Kimberly77 on 06 Jan 2023, 11:30, edited 1 time in total.
GMAT Club Legend
Joined: 03 Oct 2013
Affiliations: CrackVerbal
Posts: 4914
Own Kudos [?]: 7830 [4]
Given Kudos: 221
Location: India
If x is a positive integer, how many positive integers less than x are [#permalink]
3
Kudos
1
Bookmarks
Top Contributor
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)
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 = $$(prime number 2 )^1$$. 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.

Devmitra Sen

Connect with me on Linkedin here
Non-Human User
Joined: 09 Sep 2013
Posts: 34222
Own Kudos [?]: 857 [0]
Given Kudos: 0
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: If x is a positive integer, how many positive integers less than x are [#permalink]
Moderator:
Math Expert
94776 posts