Last visit was: 11 Sep 2024, 22:43 It is currently 11 Sep 2024, 22:43
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 the prime numbers p and t are the only prime factors of

SORT BY:
Tags:
Show Tags
Hide Tags
Intern
Joined: 23 Oct 2009
Posts: 3
Own Kudos [?]: 109 [109]
Given Kudos: 0
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657784 [52]
Given Kudos: 87242
General Discussion
Intern
Joined: 10 Dec 2009
Posts: 2
Own Kudos [?]: 1 [0]
Given Kudos: 0
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657784 [1]
Given Kudos: 87242
Re: Prime factors [#permalink]
1
Kudos
brownybuddy
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.
Intern
Joined: 12 Oct 2008
Posts: 29
Own Kudos [?]: 14 [0]
Given Kudos: 3
Re: Prime factors [#permalink]
Excellent question as well as explanation.
Manager
Joined: 07 Feb 2010
Posts: 101
Own Kudos [?]: 4110 [0]
Given Kudos: 101
Re: Prime factors [#permalink]
thanks bunuel good expalanation
Verbal Forum Moderator
Joined: 31 Jan 2010
Posts: 310
Own Kudos [?]: 352 [0]
Given Kudos: 149
Q49  V42
WE 1: 4 years Tech
Re: Prime factors [#permalink]
Ans is B , good one Bunuel
Alum
Joined: 19 Mar 2012
Posts: 4330
Own Kudos [?]: 51875 [4]
Given Kudos: 2326
Location: United States (WA)
Concentration: Leadership, General Management
Schools: Ross '20 (M)
GMAT 1: 760 Q50 V42
GMAT 2: 740 Q49 V42 (Online)
GMAT 3: 760 Q50 V42 (Online)
GPA: 3.8
WE:Marketing (Non-Profit and Government)
Re: If the prime numbers p and t are the only prime factors [#permalink]
3
Kudos
1
Bookmarks
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.
Director
Joined: 22 Mar 2011
Posts: 518
Own Kudos [?]: 2174 [0]
Given Kudos: 43
WE:Science (Education)
Re: If the prime numbers p and t are the only prime factors [#permalink]
ankit0411
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.

Originally posted by EvaJager on 25 Sep 2012, 11:49.
Last edited by EvaJager on 25 Sep 2012, 11:54, edited 1 time in total.
BSchool Moderator
Joined: 28 May 2012
Posts: 83
Own Kudos [?]: 433 [0]
Given Kudos: 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]
souvik101990
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 ?
BSchool Moderator
Joined: 28 May 2012
Posts: 83
Own Kudos [?]: 433 [0]
Given Kudos: 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]
EvaJager
ankit0411
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 ? .
Director
Joined: 22 Mar 2011
Posts: 518
Own Kudos [?]: 2174 [0]
Given Kudos: 43
WE:Science (Education)
Re: If the prime numbers p and t are the only prime factors [#permalink]
ankit0411
EvaJager
ankit0411
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.
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657784 [0]
Given Kudos: 87242
Re: If the prime numbers p and t are the only prime factors [#permalink]
ankit0411
EvaJager
ankit0411
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.
Alum
Joined: 19 Mar 2012
Posts: 4330
Own Kudos [?]: 51875 [0]
Given Kudos: 2326
Location: United States (WA)
Concentration: Leadership, General Management
Schools: Ross '20 (M)
GMAT 1: 760 Q50 V42
GMAT 2: 740 Q49 V42 (Online)
GMAT 3: 760 Q50 V42 (Online)
GPA: 3.8
WE:Marketing (Non-Profit and Government)
Re: If the prime numbers p and t are the only prime factors of [#permalink]
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.
BSchool Moderator
Joined: 28 May 2012
Posts: 83
Own Kudos [?]: 433 [0]
Given Kudos: 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]
Quote:
Check this: math-number-theory-88376.html It might help.

Thanks Bunnuel ! I have gone through that, very valuable !
Senior Manager
Joined: 13 Aug 2012
Posts: 327
Own Kudos [?]: 1857 [1]
Given Kudos: 11
Concentration: Marketing, Finance
GPA: 3.23
Re: If the prime numbers p and t are the only prime factors of [#permalink]
1
Bookmarks
phoenixgmat
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!

Director
Joined: 29 Nov 2012
Posts: 575
Own Kudos [?]: 6236 [0]
Given Kudos: 543
Re: If the prime numbers p and t are the only prime factors of [#permalink]
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 )
Math Expert
Joined: 02 Sep 2009
Posts: 95451
Own Kudos [?]: 657784 [0]
Given Kudos: 87242
Re: If the prime numbers p and t are the only prime factors of [#permalink]
fozzzy
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.
Manager
Joined: 22 Feb 2009
Posts: 109
Own Kudos [?]: 542 [0]
Given Kudos: 148
Re: If the prime numbers p and t are the only prime factors of [#permalink]
Bunuel
phoenixgmat
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
Director
Joined: 26 Oct 2016
Posts: 506
Own Kudos [?]: 3422 [1]
Given Kudos: 877
Location: United States
Concentration: Marketing, International Business
Schools: HBS '19
GMAT 1: 770 Q51 V44
GPA: 4
WE:Education (Education)
Re: If the prime numbers p and t are the only prime factors of [#permalink]
1
Kudos
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of t*p^2?
(1) m has more than 9 positive factors
(2) m is a multiple of p^3

Draw a prime box and put p and t inside. According to the problem, there could be multiple instances of p and t in there, but that's it. We want to know whether there are at least two p's and one t in there.

Start with statement 2. If m is a multiple of p^3, that means there are 3 p's in m's prime box. There's already a t in there, according to the original question. So there are at least 2 p's and one t. Answer to question is yes, so statement is sufficient. Eliminate A, C, E.

Statement 1. Notice that this just says "positive factors" NOT prime factors. The complete set of factors is made by multiplying the prime factors in different combinations. For example, 12 has the prime factors 2, 2, and 3. We can find all of the general factors of 12 by taking 2, 3, 2*2, 2*3, 2*2*3, and of course 1.

So m has more than 9 positive factors. Well, I know m has p and t - there are 2 factors. And I know m has 1 and itself - there are 2 more factors, for a total of four. I need five more, so I have to add to my prime box to be able to create five more general factors. The only things I can put in my prime box are p and t. I can put all p's, all t's, or some combination of p's and t's. If I put in at least one p, then I'd have at least 2 p's and one t, which would answer the question "yes." BUT, if I put in all t's, then I'd only have one p, which would answer the question "no" - so the statement is insufficient.

Hence B.
Re: If the prime numbers p and t are the only prime factors of [#permalink]
1   2
Moderator:
Math Expert
95451 posts