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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!