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 Your Progress

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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 350,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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.

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 10:45

1

This post received 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 ofp^2t. 2. The other seven factors could havep^2. In that case m would be a multiple of p^2t So, Insufficient. Lets look at statement 2 If m is a multiple ofp^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.

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 10:49

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.

Answer B. _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Last edited by EvaJager on 25 Sep 2012, 10:54, edited 1 time in total.

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 10:52

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 ofp^2t. 2. The other seven factors could havep^2. In that case m would be a multiple of p^2t So, Insufficient. Lets look at statement 2 If m is a multiple ofp^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 .

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 10:55

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.

Answer B.

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

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 11:00

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.

Answer B.

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.

Re: If the prime numbers p and t are the only prime factors [#permalink]
25 Sep 2012, 11:02

Expert's post

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.

Answer B.

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

Re: If the prime numbers p and t are the only prime factors of [#permalink]
22 Jan 2013, 02:21

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!

My three goals of business school: entrepreneurship, network, and professor mentor. I want to build something. I want to meet new people and create life-long friendships. I want to...