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# If m is an integer greater than zero but less than integer n

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If m is an integer greater than zero but less than integer n [#permalink]

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05 Aug 2012, 23:04
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If m is an integer greater than zero but less than integer n, is m a factor of n?

(1) n is divisible by all integers less than 10
(2) m is not a multiple of a prime number
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Apr 2014, 09:22, edited 3 times in total.
Edited the question and added the OA
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Re: If m is an integer greater than zero but less than n, [#permalink]

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06 Aug 2012, 02:11
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ananthpatri wrote:
If m is an integer greater than zero but less than n, is m a factor of n?

(1) n is divisible by all integers less than 10.
(2) m is not a multiple of a prime number.

(1) n must be divisible by 2, 3, 4,...,9 so, n is divisible by 9! If m = 11, n is not necessarily divisible by m.
Not sufficient.

(2) m must be 1, therefore m is a factor of n. Sufficient.

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Re: If m is an integer greater than zero but less than n, [#permalink]

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06 Aug 2012, 20:11
EvaJager wrote:
ananthpatri wrote:
If m is an integer greater than zero but less than n, is m a factor of n?

(1) n is divisible by all integers less than 10.
(2) m is not a multiple of a prime number.

(1) n must be divisible by 2, 3, 4,...,9 so, n is divisible by 9! If m = 11, n is not necessarily divisible by m.
Not sufficient.

(2) m must be 1, therefore m is a factor of n. Sufficient.

What does this mean ?
m is not a multiple of a prime number

I thought it means that M is a Prime Number itself.
Can m be 19 ?

How can m be 1 ? 1 is a multiple of one. ? All numbers are a multiple of one. It can then be 2 as well ? 3 as well ? 5 as well ?
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Re: If m is an integer greater than zero but less than n, [#permalink]

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06 Aug 2012, 22:12
mandyrhtdm wrote:
EvaJager wrote:
ananthpatri wrote:
If m is an integer greater than zero but less than n, is m a factor of n?

(1) n is divisible by all integers less than 10.
(2) m is not a multiple of a prime number.

(1) n must be divisible by 2, 3, 4,...,9 so, n is divisible by 9! If m = 11, n is not necessarily divisible by m.
Not sufficient.

(2) m must be 1, therefore m is a factor of n. Sufficient.

What does this mean ?
m is not a multiple of a prime number

I thought it means that M is a Prime Number itself.
Can m be 19 ?

19 = 19*1, is a multiple of a prime number 19. So, m cannot be 19.

How can m be 1 ? 1 is a multiple of one. ? All numbers are a multiple of one. It can then be 2 as well ? 3 as well ? 5 as well ?

1 is not prime. 2, 3, 5 are primes and all are a multiple of themselves. Neither one can be m.
Every positive integer greater than 1 has a unique factorization, meaning it can be written as a product of prime numbers.
1 cannot.

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Re: If m is an integer greater than zero but less than n, [#permalink]

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11 Aug 2012, 04:08
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ananthpatri wrote:
If m is an integer greater than zero but less than n, is m a factor of n?

(1) n is divisible by all integers less than 10.
(2) m is not a multiple of a prime number.

stmt 1: n = LCM(2-9) * K - INSUFFICIENT
stmt 2: m is not a multiple of prime number means,
m can not be a composite number, because every composite number can be factorized in to product of prime numbers

m can not even be a prime number, because every prime number is a multiple of itself.

m has to be an integer greater than 0. only value n can have is 1.

coming to root question. is m a factor of n. If m = 1, yes..! As 1 is a multiple of every integer. - SUFFICIENT

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Re: If m is an integer greater than zero but less than n, [#permalink]

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11 Aug 2012, 10:23
I reached the same conclusion hermit84 but then I realized that we are not explicitely told that n is an integer, a condition I think is necessary for statement 2 to be sufficient on its own.

Hence, C (if my assumption holds).
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Re: If m is an integer greater than zero but less than n, [#permalink]

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11 Aug 2012, 10:49
leodesrumaux wrote:
I reached the same conclusion hermit84 but then I realized that we are not explicitely told that n is an integer, a condition I think is necessary for statement 2 to be sufficient on its own.

Hence, C (if my assumption holds).

The question is about m being a factor of n. The term factor is used only for integers.
So, n is also an integer.

I agree, it would have been better to state clearly that both m and n are positive integers.
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Re: If m is an integer greater than zero but less than n, [#permalink]

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11 Aug 2012, 10:57
EvaJager wrote:
ananthpatri wrote:
If m is an integer greater than zero but less than n, is m a factor of n?

(1) n is divisible by all integers less than 10.
(2) m is not a multiple of a prime number.

(1) n must be divisible by 2, 3, 4,...,9 so, n is divisible by 9! If m = 11, n is not necessarily divisible by m.
Not sufficient.

(2) m must be 1, therefore m is a factor of n. Sufficient.

hermit84 is right: n must be divisible by the least common multiple of all the integers less than 10, which is not 9! as I stated in my above post.
The difference is a factor of $$2^3*3$$, so, 9! is not accurate. The rest holds, and doesn't change the conclusion.
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Re: If m is an integer greater than zero but less than n, [#permalink]

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11 Aug 2012, 12:25
leodesrumaux wrote:
I reached the same conclusion hermit84 but then I realized that we are not explicitely told that n is an integer, a condition I think is necessary for statement 2 to be sufficient on its own.

Hence, C (if my assumption holds).

leodesrumaux, be cautious but not extra cautious.
I know there are many DS questions, which will test the same thing you pointed, but not here.
n has to be integer, if we are talking about factors here.
Even if we assume that every whole number can have a factor, we will still have to accept 1 as a universal factor.

I hope it makes sense.
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Re: If m is an integer greater than zero but less than n, [#permalink]

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29 Aug 2012, 15:39
That one is interesting!

If m is an integer greater than zero but less than n, is m a factor of n?
what do we learn?
- m is an integer > 0 , m can be 1,2,3,4...................
- m must be less than n, n < m, note that we don't know whether n is an integer; we simply know that n is positive
Now the question: Is m a factor of n OR is n a multiple of m?

Before tackling the statements, let us give a few conditions that will make n divisible by m:
1- n is an integer, therefore
- m=1
- m has all the prime factors of n ( so m cannot be prime)

2- n is not an integer, therefore
n/m cannot be an integer.

What do the statements reveal or imply ?

1) n is divisible by all intergers less than 10
- n is an integer
- n is not prime (1,2,3,4........,9 are factors of n)
as you may notice, this statement is not sufficient: n is an integer, and is not prime. if m can be written as a product of prime factors less than 10, n/m would be sufficient; but m can be written as a product of prime factors less than 10 and a prime (n = 2*11 =22 or n = 3*17 = 51), and n/m will not necessarily be a integer. Not sufficient

2) m is not a multiple of a prime number
this clearly means that m cannot be written as the product of a prime integer
therefore m must be 1, since any non-prime can be written as the product of prime integers (2,3,5,7,11,13.....) AND a prime is a multiple of itself

clearly statement (2) is not sufficient
n can still be any number (integer, fraction etc.); if n were a fraction n/1 cannot be an integer. for n/m to be an integer, n must be an integer. But statement (2) does not give us enough information about n.

BUT COMBINING THE TWO STATEMENTS IS SUFFICIENT
- n is an integer and n can be written as the product of all integers from 2 to 9 (from statement 1)
- n< m (from the stem)
- m = 1 (statement 2)

Therefore C must the correct answer

Last edited by silas on 29 Aug 2012, 18:25, edited 1 time in total.
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Re: If m is an integer greater than zero but less than n, [#permalink]

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29 Aug 2012, 15:51
hermit84 wrote:
leodesrumaux wrote:
I reached the same conclusion hermit84 but then I realized that we are not explicitely told that n is an integer, a condition I think is necessary for statement 2 to be sufficient on its own.

Hence, C (if my assumption holds).

leodesrumaux, be cautious but not extra cautious.
I know there are many DS questions, which will test the same thing you pointed, but not here.
n has to be integer, if we are talking about factors here.
Even if we assume that every whole number can have a factor, we will still have to accept 1 as a universal factor.

I hope it makes sense.

@hermit84
leodesrumaux is right to be cautious or extra cautious.
We must not assume anything that is not clearly stated.
And in the question stem, it is clearly stated that m is an integer.
And they want us to decide whether a number "n" is a multiple of integer m.
Even though 1 is a universal factor, n/1 will not be an integer if n is decimal or a fraction.
And that is one the the three tricks I noticed with the question

Hope this helps
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Re: If m is an integer greater than zero but less than n, [#permalink]

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08 Apr 2014, 09:05
What's the answer for this one? I too thought that it didn't matter whether 'n' was an integer since 1 is a factor of any number whether integer or not.

Therefore, IMHO B stands

Please let us know will ya?
Thanks
Cheers
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Re: If m is an integer greater than zero but less than n, [#permalink]

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08 Apr 2014, 09:26
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Expert's post
jlgdr wrote:
What's the answer for this one? I too thought that it didn't matter whether 'n' was an integer since 1 is a factor of any number whether integer or not.

Therefore, IMHO B stands

Please let us know will ya?
Thanks
Cheers
J

Term "factor" is used only for integers. Also, the question was copied incorrectly, actual question says that n is in fact an integer.

If m is an integer greater than zero but less than integer n, is m a factor of n?

(1) n is divisible by all integers less than 10. Clearly insufficient.

(2) m is not a multiple of a prime number. Every positive integer is a multiple of at least one prime number except 1, thus m=1, which implies that it must be a factor of any other integer including n. Sufficient.

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Re: If m is an integer greater than zero but less than n,   [#permalink] 08 Apr 2014, 09:26
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