GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Nov 2019, 07:15

### 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

# If t, p, and q are different positive integers, how many positive inte

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 59017
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:00
2
15
00:00

Difficulty:

95% (hard)

Question Stats:

31% (01:38) correct 69% (01:38) wrong based on 285 sessions

### HideShow timer Statistics

If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) $$t = p*q$$; p and q have no common prime factors.
(2) $$t = p*q$$; p and q each have exactly 5 positive integer factors.

 This question was provided by Experts Global for the Game of Timers Competition

_________________
Intern
Joined: 14 Jan 2016
Posts: 20
Location: India
Concentration: Marketing, General Management
GMAT 1: 710 Q50 V35
GMAT 2: 750 Q50 V41
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:53
2
1
Quote:
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1)$$t=p∗q;$$ p and q have no common prime factors.
(2)$$t=p∗q;$$ p and q each have exactly 5 positive integer factors.

(1)$$t=p∗q;$$ p and q have no common prime factors.
We can't arrive to the number of factors of q with this information.
p can be 2 and q can be 3.
This would make t = 6, number of factors would be 4 (1,2,3 and 6)
Or p can be 4 and q can be 3
This would make t = 12, number of factors would be 6 (1,2,3,4,6 and 12)
Insufficient.

(2)$$t=p∗q;$$ p and q each have exactly 5 positive integer factors.
Let N be a number. N = $$a^x b^y c^z$$ , where a, b and c are different prime numbers.
The number of factors of N = $$(x+1)*(y+1)*(z+1)$$
Now, for a positive number to have odd number of factors, it needs to be a perfect square.
since both p and q have 5 factors and both are different (given in stem), they must be fourth powers of different primes.
let $$p = a^4$$ and $$q = b^4$$ where both a and b are distinct primes.
Now $$t = p*q = a^4 * b^4$$
So the number of factors t will have =$$(4+1) * (4+1) = 25$$
Sufficient.

Hence (B)
_________________
The only alternative to hard work is HARDER work.
##### General Discussion
Intern
Joined: 15 Jun 2019
Posts: 32
Location: Kazakhstan
Schools: Carey '21
GPA: 3.93
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:08
2
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=q∗q; p and q have no common prime factors.
This statement doesn't tell us about the number of factors of q (we don't know if q is prime)
For example, it could be that t = 12*12 or t = 13*13 etc

(2) t=p∗q; p and q each have exactly 5 positive integer factors.

We don't know if these integer factors are the same for p and q. They may be different or the same

However, (1) and (2) combined allow us to understand that those prime factors for p and q are different
Sufficient to answer the question

C
GMAT Club Legend
Joined: 18 Aug 2017
Posts: 5261
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:23
1
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=q∗qt=q∗q; p and q have no common prime factors.
(2) t=p∗qt=p∗q; p and q each have exactly 5 positive integer factors.

#1
t=q^2 ; no relation of p given insufficient
#2
t=p*q and p& q have 5 + integer factors
so p =2^4 and q=3^4 so t= 16*81 factor of t ; 25 factors
IMO B
Manager
Status: Not Applying
Joined: 27 Apr 2009
Posts: 178
Location: India
Schools: HBS '14 (A)
GMAT 1: 730 Q51 V36
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:24
1
If n = a^p * b^q * c^r....., where a, b, c are prime factors.
then the number of factors of n is (p + 1)*(q + 1)*(r + 1)......

Considering statement (1) alone:
t = q^2
But we don't know if q is prime
INSUFFICIENT

Considering statement (2) alone:
t = p * q
If p and q have exactly 5 factors, then p and q must be (prime)^4
Hence, the number of factors of t = (4 + 1)(4 + 1) = 25
SUFFICIENT
The answer is (B).
_________________
http://www.wizius.in
Better Prep. Better Scores. Better Schools

Guaranteed Admission to Top-50 MBA Programs
You either get-in or get your money-back.
Director
Joined: 28 Jul 2016
Posts: 642
Location: India
Concentration: Finance, Human Resources
GPA: 3.97
WE: Project Management (Investment Banking)
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:32
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=p∗qt=p∗q; p and q have no common prime factors.
(2) t=p∗qt=p∗q; p and q each have exactly 5 positive integer factors.

general formula for finding factors for any number of form $$a^x *b^y$$
(x+1)(y+1)
using statement 1
:
$$t = p * q$$ where they have no common factor
Now p can be $$2^3$$ and q can be 5^2
or p can be$$2^3 * 3^3$$ and q can$$5^11 * 7*11$$
Hence we cant find a single solution

Using statement 2 :
each p and q has 5 unique factors
so $$p =3^4$$ and$$q = 2^4$$ is also valid therefore # number of factors = 5*5 = 25
and $$p = 2^4$$and $$q = 2^4$$is also valid therefore # number of factors = (8+1) = 9
Not sufficient
Now combine both the statements
the only valid scenario
will be of the form
$$a^x * b^y$$ such as$$p =3^4$$ and $$q =2^4$$

Hence total 25
Manager
Joined: 29 Nov 2018
Posts: 148
Location: India
Concentration: Entrepreneurship, General Management
GPA: 3.99
WE: Engineering (Computer Hardware)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:32
Correct answer is C.
Option 1 tells us that p and q does not have any common prime factors. But p and q can be anything. for eg: p =2 and q =3 or p=4 and q = 9.
In both cases t will have different number of factors.
Hence insufficient.

Option2:
each of p and q have 5 integral factors. But we dont know if there are some overlaps.

Hence insufficient.

Option 1 and Option2:
We know factors of both p and q. Also we know p and q does not have any common prime factor. Hence number of factors of t can be found out.

Hence sufficient.

Manager
Joined: 11 Feb 2013
Posts: 216
Location: United States (TX)
GMAT 1: 490 Q44 V15
GMAT 2: 690 Q47 V38
GPA: 3.05
WE: Analyst (Commercial Banking)
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

Updated on: 04 Jul 2019, 10:40
Statement 1: INSUFFICIENT
Case 1: p=2 & q=3, So, t=6 (4 factors of t.)
Case 2: p=4 & q=9, So, t=36 (9 factors of t.)

Statement 2: INSUFFICIENT
ONLY PERFECT SQUARES HAVE ODD NUMBER OF FACTORS.
Case 1: p=2^4 & q=2^4, So, t=2^8 [ 9 factors of t.)
Case 2:p=2^4 & q=3^4, So, t=2^4*3^4 (25 factors of t)

Combining: CASE 1 of statement 2 is NOT POSSIBLE.
ONLY CASE 1 of statement 2 is NOT POSSIBLE that means t=(any integer)^8= 9 Positive factors.

C

Originally posted by BelalHossain046 on 04 Jul 2019, 08:42.
Last edited by BelalHossain046 on 04 Jul 2019, 10:40, edited 2 times in total.
Manager
Joined: 08 Jan 2018
Posts: 145
Location: India
Concentration: Operations, General Management
WE: Project Management (Manufacturing)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:45
1
Refer Attached Image.

Ans. B
Attachments

WhatsApp Image 2019-07-04 at 9.13.36 PM.jpeg [ 73.04 KiB | Viewed 2300 times ]

Intern
Joined: 25 Aug 2015
Posts: 45
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:48
(1) this means t has at least two different factors, which are p and q. However, we don't know if p or q has the value of 1. if for example, p is 1 and q is 3, then t is 3, and t has only 2 factors. If p is 2 and q is 3, then t has factors of 1, 6, 2 and 3 (four factors). So we do not know from 1.

(2) p and q have 5 positive integer factors, however we don't know if the integer factors of p and q are all different/unique numbers.

so each is not sufficient. taken together, we know that p and q are not 1, and that they each have different integer factors except for 1. hence the positive integers of t is 9. (5+5-1 as 1 will be the common integer of p and q)
SVP
Joined: 03 Jun 2019
Posts: 1837
Location: India
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:51
1
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=p∗q; p and q have no common prime factors.
No of factors of p & q are not known but it is mentioned that they have no common prime factors.
For example of p=2^3 and q =3^4, they have 4 & 5 factors respectively and no common prime factors
But if p=2 and q=3, they have 2 factors each with no common prime factors
Still it is not possible to ascertain no of positive integer factors of t based on information provided. NOT SUFFICIENT.
(2) t=p∗q; p and q each have exactly 5 positive integer factors.
If p=x^4 and q=y^4 where x is a prime number, they have both 5 positive integer factors and since p & q are different t= p*q will have 5*5 = 25 positive integer factors. SUFFICIENT.

Statement 2 alone is SUFFICIENT.

IMO B
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
Manager
Joined: 27 May 2010
Posts: 200
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

Updated on: 05 Jul 2019, 00:19
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=p∗q; p and q have no common prime factors.
We do not know the number of prime factors of p and q. So, not sufficient.

(2) t=p∗q; p and q each have exactly 5 positive integer factors.
We are provided with the number of factors for p and q. Odd number of factors implies both p and q are square numbers.
But no information on common factors between the two. So not sufficient.

Taking both statements together,
Both statements together are sufficient.

Option C would be the answer.

Posted from my mobile device
_________________
Please give Kudos if you like the post

Originally posted by prashanths on 04 Jul 2019, 08:51.
Last edited by prashanths on 05 Jul 2019, 00:19, edited 1 time in total.
Manager
Joined: 27 Feb 2017
Posts: 117
Location: United States (WA)
GMAT 1: 760 Q50 V42
GRE 1: Q169 V168
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 08:58
I spent 1 minute and 46 seconds. And my answer is (C).

(1) t=p∗q; p and q have no common prime factors. As we do not how many factors p or q have, there is no way to determine how many factors t has.
(2) t=p∗q; p and q each have exactly 5 positive integer factors. Here: If p and q are the same number, we can surely determine how many factors t has (no need to calculate, but the answer is 9.) But if P and q are not the same number, the factors for t will be different.

(1) and (2) together: We can tell that p and q are definitely not the same. Then, we have 2 approaches.
(a) If p and q do not share any common prime factors, they do not share other factors except for 1. That fact alone should enable us to determine that we can figure out how many factors t has.
(b) it turns out that p and q can only be expressed as primeNumber^4 (such as 16, or 27), we can determine that t has a total of 25 factors.
Sufficient.
Intern
Joined: 29 May 2019
Posts: 32
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

Updated on: 04 Jul 2019, 09:15
A is the correct answer

P*q can determine the factors when co prime

B will not provide unique solutions

Posted from my mobile device

Originally posted by Kssss on 04 Jul 2019, 08:58.
Last edited by Kssss on 04 Jul 2019, 09:15, edited 1 time in total.
Director
Joined: 22 Nov 2018
Posts: 562
Location: India
GMAT 1: 640 Q45 V35
GMAT 2: 660 Q48 V33
If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

Updated on: 05 Jul 2019, 09:51
(1) t=q∗q; p and q have no common prime factors. - means the number will be in the form q^2 but we are not aware of the factors of q and p (Just mentioned they are co prime i.e. 64 and 65) so they can have number of factors which will affect the total number of factors of the number t - Insufficient

(2) t=p∗q; p and q each have exactly 5 positive integer factors.- Provides that P and Q are perfect Squares. No information on common factors Insufficient

Combined - sufficient p=16 and Q=625 or p=625 and q=81 a^4*b^4 were a and b are prime so 25 factors

IMO C
_________________
Give +1 kudos if this answer helps..!!

Originally posted by Arvind42 on 04 Jul 2019, 08:59.
Last edited by Arvind42 on 05 Jul 2019, 09:51, edited 1 time in total.
Senior Manager
Joined: 12 Dec 2015
Posts: 437
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:14
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=p∗q; p and q have no common prime factors. --> can't say how many factors of t(p=2 & q=3 or p=2*5 & q = 3*7)
(2) t=p∗q; p and q each have exactly 5 positive integer factors. --> can't say because there can be common factors between p & q

combining (1) & (2):
we can say p & q have 5 factor each & there is no common factor between them, so t have finite number of common factor.
So the answer is C
Manager
Joined: 30 Nov 2017
Posts: 193
WE: Consulting (Consulting)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:22
1
Statement 1 is not sufficient as we do not know the value of p and q.

Statement 2 says that p and q have exactly 5 factors
Example is (x^2)^2 = x^4
Such as 2^4 = 16
Factors of 16 = 1,2,4,8,16
The same thing with 3^4 = 81
Factors of 81 = 1,3,9,27,81
Thus, this statement is sufficient, hence answer choice "B"

Posted from my mobile device
_________________
Be Braver, you cannot cross a chasm in two small jumps...
Manager
Joined: 05 Feb 2016
Posts: 166
Location: India
Concentration: General Management, Marketing
WE: Information Technology (Computer Software)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:29
from:1 t=p∗q; p and q have no common prime factors.

not sufficient, since p=2,q=3 and p=2,q=9

from: 2 t=p∗q; p and q each have exactly 5 positive integer factors
Not sufficient

from both 1 and 2

Not sufficient
E
Manager
Joined: 28 Feb 2014
Posts: 178
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE: Engineering (Education)
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:29
1
1
(1) t=p∗q; p and q have no common prime factors.
p and q are coprimes, let p=2, q=3 then no. of factors of t are 4
When p=6 and q=7; no. of factors of t are 8. Insufficient.

(2) t=p∗q; p and q each have exactly 5 positive integer factors.
As p and q are different positive integers and each have 5 factors, then number of factors of t are 25. Sufficient.

B is correct.
Manager
Joined: 17 Jan 2017
Posts: 87
Re: If t, p, and q are different positive integers, how many positive inte  [#permalink]

### Show Tags

04 Jul 2019, 09:30
1
If t, p, and q are different positive integers, how many positive integers are factors of t?

(1) t=p∗q; p and q have no common prime factors.
(2) t=p∗q; p and q each have exactly 5 positive integer factors.

The question needs the value of t to determine what is the number of positive integers as factors.

Stmt 1: p and q don't share any common prime factors in which t is the multiplication of p and q. but, there are many results possible for p and q. so, factors of t will also vary. hence, insufficient. Eliminate A, D.

Stmt 2: it states that t is the multiplication of p and q and also that the definite number of factors of both p and q are 5.
this is sufficient.
So, the correct answer choice is (B)
Re: If t, p, and q are different positive integers, how many positive inte   [#permalink] 04 Jul 2019, 09:30

Go to page    1   2   3   4    Next  [ 66 posts ]

Display posts from previous: Sort by

# If t, p, and q are different positive integers, how many positive inte

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne