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

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If the prime numbers p and t are the only prime factors of [#permalink] New post 26 Oct 2009, 13:53
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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 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, 03:04, edited 1 time in total.
Edited the question and added the OA
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Re: Prime factors [#permalink] New post 26 Oct 2009, 15:16
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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.

Answer: B.
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Re: Prime factors [#permalink] New post 11 Dec 2009, 02:06
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Re: if the prime no.s p & t [#permalink] New post 16 Nov 2010, 06:05
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 25 Sep 2012, 10:45
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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.
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Re: Prime factors [#permalink] New post 10 Dec 2009, 20:19
Excellent explanation but are we assuming that 'p' and 't' are different prime factors i.e. 'p' is not equal to 't'?
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Re: Prime factors [#permalink] New post 11 Dec 2009, 19:07
Excellent question as well as explanation.
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Re: Prime factors [#permalink] New post 17 Nov 2010, 07:48
thanks bunuel good expalanation
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Re: Prime factors [#permalink] New post 20 Nov 2010, 05:31
Ans is B , good one Bunuel
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 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.
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Last edited by EvaJager on 25 Sep 2012, 10:54, edited 1 time in total.
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 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 .

Is my thinking right ?
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 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 ? .
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 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.
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 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 ? .


Check this: math-number-theory-88376.html It might help.
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Re: If the prime numbers p and t are the only prime factors of [#permalink] New post 25 Sep 2012, 11:16
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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.
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Re: If the prime numbers p and t are the only prime factors [#permalink] New post 25 Sep 2012, 19:39
Quote:
Check this: math-number-theory-88376.html It might help.


Thanks Bunnuel ! I have gone through that, very valuable !
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Re: If the prime numbers p and t are the only prime factors of [#permalink] New post 25 Sep 2012, 19:40
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.
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Re: If the prime numbers p and t are the only prime factors of [#permalink] New post 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!

SUFFICIENT!

ANswer: B
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Re: If the prime numbers p and t are the only prime factors of [#permalink] New post 16 Jul 2013, 07:10
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 )
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Re: If the prime numbers p and t are the only prime factors of [#permalink] New post 16 Jul 2013, 07:41
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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.
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Re: If the prime numbers p and t are the only prime factors of   [#permalink] 16 Jul 2013, 07:41
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