Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 10 Dec 2016
Posts: 12

If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
Updated on: 01 Mar 2018, 01:10
Question Stats:
45% (02:26) correct 55% (02:19) wrong based on 131 sessions
HideShow timer Statistics
If P is a prime number and Q is a positive integer, how many factors does P^2*Q have? (1) The lowest number that has both P^2 and Q^3 as its factors is 5400 (2) P and Q have only one common factor
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by sekharm2389 on 01 Mar 2018, 00:53.
Last edited by Bunuel on 01 Mar 2018, 01:10, edited 1 time in total.
Renamed the topic and edited the question.



Math Expert
Joined: 02 Sep 2009
Posts: 50042

If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
01 Mar 2018, 01:25
If P is a prime number and Q is a positive integer, how many factors does P^2*Q have?(1) The lowest number that has both P^2 and Q^3 as its factors is 5400. This means that 5,400 is the least common multiple of P^2 (square of a prime) and Q^3 (perfect cube), so both P^2 and Q^3 are factors of 5,400. Factorize: \(5400 = 2^3*3^3*5^2\). P^2, which is a square of a prime, must be 5^2 (so P must be 5) because in any other case (say if P^2 is 2^2 or 3^2), the remaining multiple will not be a perfect square (Q^3). Therefore, \(P^2 = 5^2\) (P = 5) and \(Q^3 = 2^3*3^3\) (Q = 6). \(P^2*Q = 5^2*6 = 5^2*2*3\). The number of factors \(= (2 + 1)(1 + 1)(1 + 1) = 12\). Sufficient. (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ... Answer: A. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 04 Feb 2016
Posts: 72
Location: India
Concentration: Technology, Marketing
GPA: 4
WE: Sales (Computer Software)

If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 09:15
Hi Bunuel, Why are we assuming P^2 and Q^3 are the only factors apart from 1 for the number 5400? I considered multiple possibilities for Q [2,3,6,1] so went with E. I think I am missing something in the reasoning. Can you help? Tx.
_________________
GMAT1 650 Q48 V32.



Math Expert
Joined: 02 Sep 2009
Posts: 50042

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 10:01
srinjoy1990 wrote: Hi Bunuel,
Why are we assuming P^2 and Q^3 are the only factors apart from 1 for the number 5400?
I considered multiple possibilities for Q [2,3,6,1] so went with E. I think I am missing something in the reasoning. Can you help?
Tx. How can P^2 and Q^3 be the only factors of 5,400? A number to have four factors should be of the form prime1*prime2, in this case its factors would be 1, p1, p2, and p1*p2. If you say that Q can 1, 2, or 3, then what would P and would 5,400 be the LCM of P^2 an Q^3?
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Director
Joined: 30 Jan 2016
Posts: 756
Location: United States (MA)

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 11:03
Quote: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ... Are 2 prime numbers coprime? 2 and 7, for example. Thanks
_________________
Non progredi est regredi



Director
Joined: 30 Jan 2016
Posts: 756
Location: United States (MA)

If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 11:06
Akela wrote: Quote: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ... Are 2 prime numbers coprime? 2 and 7, for example. Thanks Oh, I found the answer. Thanks! Two integers a and b are said to be coprime if the only positive integer that evenly divides both of them is 1. That is, the only common positive factor of the two numbers is 1. So, according to the definition any two distinct primes are coprime.
_________________
Non progredi est regredi



Math Expert
Joined: 02 Sep 2009
Posts: 50042

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 11:07
Akela wrote: Quote: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ... Are 2 prime numbers coprime? 2 and 7, for example. Thanks Two numbers are coprime if they do not share any common factor but 1. So, any two different primes are coprime. For example, 2 and 7 do not share any common factor but 1.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 04 Feb 2016
Posts: 72
Location: India
Concentration: Technology, Marketing
GPA: 4
WE: Sales (Computer Software)

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
09 Mar 2018, 11:52
Bunuel wrote: srinjoy1990 wrote: Hi Bunuel,
Why are we assuming P^2 and Q^3 are the only factors apart from 1 for the number 5400?
I considered multiple possibilities for Q [2,3,6,1] so went with E. I think I am missing something in the reasoning. Can you help?
Tx. How can P^2 and Q^3 be the only factors of 5,400? A number to have four factors should be of the form prime1*prime2, in this case its factors would be 1, p1, p2, and p1*p2. If you say that Q can 1, 2, or 3, then what would P and would 5,400 be the LCM of P^2 an Q^3? Thanks got my error. The only possible combination is 6^3*5^2 because, the numbers have to be coprime [The composite number cannot contain the prime number here], so the LCM is effectively product of the two. Should have noticed.
_________________
GMAT1 650 Q48 V32.



Intern
Joined: 04 Jan 2015
Posts: 3

If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
10 Mar 2018, 11:33
Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything.



Math Expert
Joined: 02 Sep 2009
Posts: 50042

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
10 Mar 2018, 11:48
shu123 wrote: Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything. First of all, please reread this: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ...Next, try to pick P and Q so that they are coprime but so that both are not primes. This should help.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 04 Jan 2015
Posts: 3

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
10 Mar 2018, 11:52
Oh ok..Thanks if i pick 7 and 9 ,no. of factors will vary.



Manager
Joined: 01 Aug 2017
Posts: 188
Location: India
Concentration: General Management, Leadership
GPA: 3.4
WE: Information Technology (Computer Software)

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
13 May 2018, 02:00
If P is a prime number and Q is a positive integer, how many factors does P^2*Q have? Statement 1: (1) The lowest number that has both P^2 and Q^3 as its factors is 5400.
This means that 5,400 is the LCM of P^2 (square of a prime) and Q^3 (perfect cube). So both P^2 and Q^3 are factors of 5,400. Factorize: \(5400 = 2^3*3^3*5^2\). P^2, which is a square of a prime, must be 5^2 (so P must be 5) Therefore, \(P^2 = 5^2\) (P = 5) And \(Q^3 = 2^3*3^3\) (Q = 6).
\(P^2*Q = 5^2*6 = 5^2*2*3\). The number of factors \(= (2 + 1)(1 + 1)(1 + 1) = 12\).
Sufficient.
Statement 2 : P and Q have only one common factor. This means that P and Q are coprime,. For example, P could be 2. Q could be 3, 3^2, 3^3, ... There is no data provided for Q. This is clearly insufficient.
Answer: A.
_________________
If it helps you please press Kudos!
Thank You Sudhanshu



Intern
Joined: 12 Sep 2017
Posts: 28

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
29 May 2018, 07:22
Bunuel wrote: shu123 wrote: Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything. First of all, please reread this: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ...Next, try to pick P and Q so that they are coprime but so that both are not primes. This should help. Hi Bunuel Are 5 and 12 are coprimes ? Thanks



Math Expert
Joined: 02 Sep 2009
Posts: 50042

Re: If P is a prime number and Q is a positive integer, how many factors
[#permalink]
Show Tags
29 May 2018, 07:30
@s wrote: Bunuel wrote: shu123 wrote: Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything. First of all, please reread this: (2) P and Q have only one common factor. This means that P and Q are coprime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ...Next, try to pick P and Q so that they are coprime but so that both are not primes. This should help. Hi Bunuel Are 5 and 12 are coprimes ? Thanks Yes. The factors of 5 are 1 and 5. The factors of 12 are 1, 2, 3, 4, 6 and 12. As you can see the only factor they share is 1, so 5 and 12 are coprime.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Re: If P is a prime number and Q is a positive integer, how many factors &nbs
[#permalink]
29 May 2018, 07:30






