It is currently 10 Dec 2017, 19:13

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If the prime numbers p and t are the only prime factors of

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

### Hide Tags

Intern
Joined: 23 Oct 2009
Posts: 3

Kudos [?]: 23 [3], given: 0

If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

26 Oct 2009, 14:53
3
KUDOS
21
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

59% (01:18) correct 41% (01:42) wrong based on 586 sessions

### HideShow timer Statistics

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2*t?

(1) m has more than 9 positive factors.
(2) m is a multiple of p^3
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Mar 2012, 04:04, edited 1 time in total.
Edited the question and added the OA

Kudos [?]: 23 [3], given: 0

Math Expert
Joined: 02 Sep 2009
Posts: 42529

Kudos [?]: 135163 [8], given: 12664

Re: Prime factors [#permalink]

### Show Tags

26 Oct 2009, 16:16
8
KUDOS
Expert's post
21
This post was
BOOKMARKED
phoenixgmat wrote:
I would appreciate some help with:

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p²t?
1) m has more than 9 positive factors.
2) m is a multiple of p³

some explanations to both statements would be great!
thx a lot

We are told that $$p$$ and $$t$$ are the ONLY prime factors of m. It could be expressed as $$m=p^x*t^y$$, where $$x$$ and $$y$$ are integers $$\geq{1}$$.

Question: is $$m$$ a multiple of $$p^2*t$$. We already know that $$p$$ and $$t$$ are the factors of $$m$$, so basically question asks whether the power of $$p$$, in our prime factorization denoted as $$x$$, more than or equal to 2: so is $$x\geq{2}$$.

(1) m has more than 9 positive factors:

Formula for counting the number of distinct factors of integer $$x$$ expressed by prime factorization as: $$n=a^x*b^y*c^z$$, is $$(x+1)(y+1)(z+1)$$. This also includes the factors 1 and $$n$$ itself.

We are told that $$(x+1)(y+1)>9$$ (as we know that $$m$$ is expressed as $$m=p^x*t^y$$)
But it's not sufficient to determine whether $$x\geq{2}$$. ($$x$$ can be 1 and $$y\geq{4}$$ and we would have their product $$>9$$, e.g. $$(1+1)(4+1)=10$$.) Not sufficient.

(2) m is a multiple of p^3
This statement clearly gives us the value of power of $$p$$, which is 3, $$x=3>2$$. So $$m$$ is a multiple of $$p^2t$$. Sufficient.

_________________

Kudos [?]: 135163 [8], given: 12664

Intern
Joined: 10 Dec 2009
Posts: 5

Kudos [?]: [0], given: 0

Re: Prime factors [#permalink]

### Show Tags

10 Dec 2009, 21:19
Excellent explanation but are we assuming that 'p' and 't' are different prime factors i.e. 'p' is not equal to 't'?

Kudos [?]: [0], given: 0

Math Expert
Joined: 02 Sep 2009
Posts: 42529

Kudos [?]: 135163 [1], given: 12664

Re: Prime factors [#permalink]

### Show Tags

11 Dec 2009, 03:06
1
KUDOS
Expert's post
brownybuddy wrote:
Excellent explanation but are we assuming that 'p' and 't' are different prime factors i.e. 'p' is not equal to 't'?

Yes. I think from the stem we can get this.
_________________

Kudos [?]: 135163 [1], given: 12664

Manager
Joined: 12 Oct 2008
Posts: 56

Kudos [?]: 2 [0], given: 3

Re: Prime factors [#permalink]

### Show Tags

11 Dec 2009, 20:07
Excellent question as well as explanation.

Kudos [?]: 2 [0], given: 3

Manager
Joined: 07 Feb 2010
Posts: 155

Kudos [?]: 776 [0], given: 101

Re: Prime factors [#permalink]

### Show Tags

17 Nov 2010, 08:48
thanks bunuel good expalanation

Kudos [?]: 776 [0], given: 101

Verbal Forum Moderator
Joined: 31 Jan 2010
Posts: 485

Kudos [?]: 163 [0], given: 149

WE 1: 4 years Tech
Re: Prime factors [#permalink]

### Show Tags

20 Nov 2010, 06:31
Ans is B , good one Bunuel
_________________

My Post Invites Discussions not answers
Try to give back something to the Forum.I want your explanations, right now !
Please let me know your opinion about the Chandigarh Gmat Centrehttp://gmatclub.com/forum/gmat-experience-at-chandigarh-india-centre-111830.html

Kudos [?]: 163 [0], given: 149

MBA Section Director
Joined: 19 Mar 2012
Posts: 4725

Kudos [?]: 18006 [1], given: 1988

Location: India
GMAT 1: 760 Q50 V42
GPA: 3.8
WE: Marketing (Non-Profit and Government)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 11:45
1
KUDOS
Expert's post
Hey
Lets look at statement 1
m has more than 9 factors
Now if p and t are the only prime factors then the other factors would be a combination of p and t either with each other or with themselves.
Now among those 9 factors, the following 2 things could happen.
1. 2 factors would be 1 and m. The other factors could be $$p, t, t^2, t^3, t^4, t^5, t^6$$. In this case the integer m is NOT a multiple of$$p^2t$$.
2. The other seven factors could have$$p^2$$. In that case m would be a multiple of $$p^2t$$
So, Insufficient.
Lets look at statement 2
If m is a multiple of$$p^3$$, then m must be a multiple of $$p^2$$. We know that m is already a multiple of t. So m must be a multiple of $$p^2t$$.
Hence Sufficient.

Hope this helps.
_________________

Kudos [?]: 18006 [1], given: 1988

Director
Joined: 22 Mar 2011
Posts: 610

Kudos [?]: 1089 [0], given: 43

WE: Science (Education)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 11:49
ankit0411 wrote:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2 t?

(1) m has more than 9 positive factors (2) m is a multiple of p^3

We can write $$m=p^a\cdot{t^b}$$ for some positive integers $$a$$ and $$b.$$

(1) The number of positive factors of $$m$$ is $$(a+1)(b+1)>9.$$
If $$a=1$$ and $$b>3$$ then $$m=pt^b$$ is not a multiple of $$p^2t.$$
If $$a>1$$ then the answer is yes.
Not sufficient.

(2) Obviously sufficient.

_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Last edited by EvaJager on 25 Sep 2012, 11:54, edited 1 time in total.

Kudos [?]: 1089 [0], given: 43

Joined: 28 May 2012
Posts: 135

Kudos [?]: 74 [0], given: 11

Location: India
Concentration: General Management, Strategy
GPA: 3.33
WE: Information Technology (Retail)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 11:52
souvik101990 wrote:
Hey
Lets look at statement 1
m has more than 9 factors
Now if p and t are the only prime factors then the other factors would be a combination of p and t either with each other or with themselves.
Now among those 9 factors, the following 2 things could happen.
1. 2 factors would be 1 and m. The other factors could be $$p, t, t^2, t^3, t^4, t^5, t^6$$. In this case the integer m is NOT a multiple of$$p^2t$$.
2. The other seven factors could have$$p^2$$. In that case m would be a multiple of $$p^2t$$
So, Insufficient.
Lets look at statement 2
If m is a multiple of$$p^3$$, then m must be a multiple of $$p^2$$. We know that m is already a multiple of t. So m must be a multiple of $$p^2t$$.
Hence Sufficient.

Hope this helps.

I got your second statement, but somehow I am not able to get the 1st statement.

For example you have p=2 and t=3, two prime numbers . Now the other 7 numbers can be any positive integer right ? i.e 4,6,8,9,4,6,8 isnt it ?

And the second case maybe that we have other 7 factors that include 2 and 3 as well . ex. 2,3,2,3,2,3,3,2,2 . In this case m is a multiple of p^2*t .

Is my thinking right ?
_________________

You want something, go get it . Period !

Kudos [?]: 74 [0], given: 11

Joined: 28 May 2012
Posts: 135

Kudos [?]: 74 [0], given: 11

Location: India
Concentration: General Management, Strategy
GPA: 3.33
WE: Information Technology (Retail)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 11:55
EvaJager wrote:
ankit0411 wrote:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2 t?

(1) m has more than 9 positive factors (2) m is a multiple of p^3

We can write $$m=p^a\cdot{t^b}$$ for some positive integers $$a$$ and $$b.$$

(1) The number of positive factors of $$m$$ is $$(a+1)(b+1)>9.$$
If $$a=1$$ and $$b>3$$ then $$m=pt^b$$ is not a multiple of $$p^2t.$$
If $$a>1$$ then the answer is yes.
Not sufficient.

(2) Obviously sufficient.

The formula you've written - (a+1)(b+1) is for the no of prime factors of a number right ? .
_________________

You want something, go get it . Period !

Kudos [?]: 74 [0], given: 11

Director
Joined: 22 Mar 2011
Posts: 610

Kudos [?]: 1089 [0], given: 43

WE: Science (Education)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 12:00
ankit0411 wrote:
EvaJager wrote:
ankit0411 wrote:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2 t?

(1) m has more than 9 positive factors (2) m is a multiple of p^3

We can write $$m=p^a\cdot{t^b}$$ for some positive integers $$a$$ and $$b.$$

(1) The number of positive factors of $$m$$ is $$(a+1)(b+1)>9.$$
If $$a=1$$ and $$b>3$$ then $$m=pt^b$$ is not a multiple of $$p^2t.$$
If $$a>1$$ then the answer is yes.
Not sufficient.

(2) Obviously sufficient.

The formula you've written - (a+1)(b+1) is for the no of prime factors of a number right ? .

NO. It is for all the positive factors of the number, including 1 and the number itself, not only prime factors.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Kudos [?]: 1089 [0], given: 43

Math Expert
Joined: 02 Sep 2009
Posts: 42529

Kudos [?]: 135163 [0], given: 12664

Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 12:02
ankit0411 wrote:
EvaJager wrote:
ankit0411 wrote:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2 t?

(1) m has more than 9 positive factors (2) m is a multiple of p^3

We can write $$m=p^a\cdot{t^b}$$ for some positive integers $$a$$ and $$b.$$

(1) The number of positive factors of $$m$$ is $$(a+1)(b+1)>9.$$
If $$a=1$$ and $$b>3$$ then $$m=pt^b$$ is not a multiple of $$p^2t.$$
If $$a>1$$ then the answer is yes.
Not sufficient.

(2) Obviously sufficient.

The formula you've written - (a+1)(b+1) is for the no of prime factors of a number right ? .

Check this: math-number-theory-88376.html It might help.
_________________

Kudos [?]: 135163 [0], given: 12664

MBA Section Director
Joined: 19 Mar 2012
Posts: 4725

Kudos [?]: 18006 [0], given: 1988

Location: India
GMAT 1: 760 Q50 V42
GPA: 3.8
WE: Marketing (Non-Profit and Government)
Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

25 Sep 2012, 12:16
Quote:
For example you have p=2 and t=3, two prime numbers . Now the other 7 numbers can be any positive integer right ? i.e 4,6,8,9,4,6,8 isnt it ?

Note that these factors are combinations of powers of the prime factors only.
_________________

Kudos [?]: 18006 [0], given: 1988

Joined: 28 May 2012
Posts: 135

Kudos [?]: 74 [0], given: 11

Location: India
Concentration: General Management, Strategy
GPA: 3.33
WE: Information Technology (Retail)
Re: If the prime numbers p and t are the only prime factors [#permalink]

### Show Tags

25 Sep 2012, 20:39
Quote:
Check this: math-number-theory-88376.html It might help.

Thanks Bunnuel ! I have gone through that, very valuable !
_________________

You want something, go get it . Period !

Kudos [?]: 74 [0], given: 11

Joined: 28 May 2012
Posts: 135

Kudos [?]: 74 [0], given: 11

Location: India
Concentration: General Management, Strategy
GPA: 3.33
WE: Information Technology (Retail)
Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

25 Sep 2012, 20:40
souvik101990 wrote:
Quote:
For example you have p=2 and t=3, two prime numbers . Now the other 7 numbers can be any positive integer right ? i.e 4,6,8,9,4,6,8 isnt it ?

Note that these factors are combinations of powers of the prime factors only.

Thanks, got that . Took me a little while to understand the solution.
_________________

You want something, go get it . Period !

Kudos [?]: 74 [0], given: 11

Senior Manager
Joined: 13 Aug 2012
Posts: 457

Kudos [?]: 568 [0], given: 11

Concentration: Marketing, Finance
GPA: 3.23
Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

22 Jan 2013, 03:21
phoenixgmat wrote:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2*t?

(1) m has more than 9 positive factors.
(2) m is a multiple of p^3

m = p^x * t^y where x is at least 1 and y is at least 1...
For m to be a multiple of p^2 * t then m must have at least 2 p and at least 1 t...

1. m has more than 9 factors
If m = p^1 * t^4 => number of factors = (1+1)(4+1) = 10 NOT A MULTIPLE!
If m = p^2 * t^3 => numbr of factors = (2+1)(3+1) = 12 A MULTIPLE!
INSUFFICIENT!

2. m is a multiple of p^3
Is it at least 2 factors of p? According to Statement (2) - YES!
Is it at least 1 factor of t? According to GIVEN - YES!

SUFFICIENT!

_________________

Impossible is nothing to God.

Kudos [?]: 568 [0], given: 11

Director
Joined: 29 Nov 2012
Posts: 863

Kudos [?]: 1484 [0], given: 543

Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

16 Jul 2013, 08:10
IS my translation for this problem correct the given info...

we know that $$\frac{m}{p*t}$$ = Integer since p and t are different integers

The question is now framed as is$$\frac{M}{p^2 t}$$ ?( T is irrelevant for this question )
_________________

Click +1 Kudos if my post helped...

Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/

GMAT Prep software What if scenarios http://gmatclub.com/forum/gmat-prep-software-analysis-and-what-if-scenarios-146146.html

Kudos [?]: 1484 [0], given: 543

Math Expert
Joined: 02 Sep 2009
Posts: 42529

Kudos [?]: 135163 [0], given: 12664

Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

16 Jul 2013, 08:41
fozzzy wrote:
IS my translation for this problem correct the given info...

we know that $$\frac{m}{p*t}$$ = Integer since p and t are different integers

The question is now framed as is$$\frac{M}{p^2 t}$$ ?( T is irrelevant for this question )

Yes, the question asks whether m/(p^2t)=integer, while saying that m/(pt)=integer.
_________________

Kudos [?]: 135163 [0], given: 12664

Manager
Joined: 22 Feb 2009
Posts: 207

Kudos [?]: 182 [0], given: 148

Re: If the prime numbers p and t are the only prime factors of [#permalink]

### Show Tags

17 Aug 2014, 15:39
Bunuel wrote:
phoenixgmat wrote:
I would appreciate some help with:

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p²t?
1) m has more than 9 positive factors.
2) m is a multiple of p³

some explanations to both statements would be great!
thx a lot

We are told that $$p$$ and $$t$$ are the ONLY prime factors of m. It could be expressed as $$m=p^x*t^y$$, where $$x$$ and $$y$$ are integers $$\geq{1}$$.

Question: is $$m$$ a multiple of $$p^2*t$$. We already know that $$p$$ and $$t$$ are the factors of $$m$$, so basically question asks whether the power of $$p$$, in our prime factorization denoted as $$x$$, more than or equal to 2: so is $$x\geq{2}$$.

(1) m has more than 9 positive factors:

Formula for counting the number of distinct factors of integer $$x$$ expressed by prime factorization as: $$n=a^x*b^y*c^z$$, is $$(x+1)(y+1)(z+1)$$. This also includes the factors 1 and $$n$$ itself.

We are told that $$(x+1)(y+1)>9$$ (as we know that $$m$$ is expressed as $$m=p^x*t^y$$)
But it's not sufficient to determine whether $$x\geq{2}$$. ($$x$$ can be 1 and $$y\geq{4}$$ and we would have their product $$>9$$, e.g. $$(1+1)(4+1)=10$$.) Not sufficient.

(2) m is a multiple of p^3
This statement clearly gives us the value of power of $$p$$, which is 3, $$x=3>2$$. So $$m$$ is a multiple of $$p^2t$$. Sufficient.

I could answer the question in 1.5 min. But I have never known the formula for counting the number of distinct factors of a integer. Thanks a ton, Bunuel
_________________

.........................................................................
+1 Kudos please, if you like my post

Kudos [?]: 182 [0], given: 148

Re: If the prime numbers p and t are the only prime factors of   [#permalink] 17 Aug 2014, 15:39

Go to page    1   2    Next  [ 26 posts ]

Display posts from previous: Sort by

# If the prime numbers p and t are the only prime factors of

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

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.