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If n is an integer, then n is divisible by how many positive

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If n is an integer, then n is divisible by how many positive  [#permalink]

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24 Dec 2013, 02:48
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.

Data Sufficiency
Question: 3
Category: Arithmetic Properties of numbers
Page: 153
Difficulty: 650

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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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24 Dec 2013, 02:49
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5
SOLUTION

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.
For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers --> n=ab, where a and b are primes, so # of factors is (1+1)(1+1)=4. Sufficient.

(2) n and 2^3 are each divisible by the same number of positive integers --> 2^3 has 4 different positive factors (1, 2, 4, and 8) so n has also 4. Sufficient.

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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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25 Dec 2013, 11:06
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Statement 1: If n is a product of two prime number, then it must be divisible by 4 positive integers. Sufficient.

Statement 2: 2^3 is divisible by 4 positive integers and since n and 2^3 are divisible by same number of positive integers, this statement is sufficient as well.

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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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26 Dec 2013, 21:52
IMO - A
To find the # of positive integers that divide n, we need to know the primes in n
(1) 2 primes For e.g. 3 and 5 so the n is 15 and divisible by 4 positive int (1,3,5 & 15) so suff.
(2) Not sure about statement 2
For e.g. 2^3 is 8.Now, if n is also 8 then sufficient.
But if n is 16 then even though both 2^3 and 16 are divisible by 4 positive numbers.n has one more positive int 16.so insuff.

Pls correct my understanding of stmt 2.Thanks
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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27 Dec 2013, 18:36
1
himapm1l wrote:
IMO - A
To find the # of positive integers that divide n, we need to know the primes in n
(1) 2 primes For e.g. 3 and 5 so the n is 15 and divisible by 4 positive int (1,3,5 & 15) so suff.
(2) Not sure about statement 2
For e.g. 2^3 is 8.Now, if n is also 8 then sufficient.
But if n is 16 then even though both 2^3 and 16 are divisible by 4 positive numbers.n has one more positive int 16.so insuff.

Pls correct my understanding of stmt 2.Thanks

It should be D mate.... Statement 2 is also sufficient. More information can be found here: math-number-theory-88376.html
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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30 Dec 2013, 02:11
If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.

Sol:
Given n is an integer and so n is divisible by how many positive integers or how many factors does n have.

Using st 1, we have n is the product of 2 different prime numbers a and b
so n =ab....let a =2 b =3 then n =6 = 2^1 *3^1 and no. of factors of n are 4 ( 1,2,3 and 6)

Consider n is of the form n = a^2*b then n= 12 and no. of factors are 6 (1,2,3,4,6 and 12 )
St 1 is not sufficient as n is a multiple of 2 prime nos. but to what powers are the prime nos. raised, we don't know. Hence A and D are ruled out

In general, when a no. p can be represented by the form p= q^a*r^b*z^c where q,r and z are prime factors raised to the powers a, b and c respectively then the number of factors will be (a+1)*(b+1)*(c+1)

St 2 says n and 2^3 ie 8 have same factors which is 4. Hence no. of factors of n are 4

Ans B.
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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02 Jun 2014, 13:56
Hi Bunuel, In statement-1, how do we know that the powers of two prime numbers is 1? I interpreted st-1 as n is a product of two different primes but their powers could be anything. So n could be ab, a^1*b, or a*b^1. Not Sufficient.
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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03 Jun 2014, 07:43
2
MensaNumber wrote:
Hi Bunuel, In statement-1, how do we know that the powers of two prime numbers is 1? I interpreted st-1 as n is a product of two different primes but their powers could be anything. So n could be ab, a^1*b, or a*b^1. Not Sufficient.

Well, first of all $$a^1*b=a*b^1=ab$$.

As for the powers: ask yourself can we says that 12=2^2*3 is the product of two different prime numbers? No.

n is the product of two different prime numbers means n = (prime 1)*(prime 2).

Hope it's clear.
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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03 Jun 2014, 07:50
Bunuel wrote:
Well, first of all $$a^1*b=a*b^1=ab$$.

Haha! Off course. Thats why silly errors are killing me.

Bunuel wrote:
As for the powers: ask yourself can we says that 12=2^2*3 is the product of two different prime numbers? No.

n is the product of two different prime numbers means n = (prime 1)*(prime 2).

Hope it's clear.

Yes makes sense. thanks!
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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28 Nov 2016, 03:39
D:

1. n = 1. p1 Xp2 so 3 . Sufficient
2. n and 2^3 implies, 2X2X2 so 3. sufficient
hence D

Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.

Data Sufficiency
Question: 3
Category: Arithmetic Properties of numbers
Page: 153
Difficulty: 650

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!
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If n is an integer, then n is divisible by how many positive  [#permalink]

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09 Jun 2017, 10:26
Bunuel wrote:
SOLUTION

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.
For more on number properties check: http://gmatclub.com/forum/math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers --> n=ab, where a and b are primes, so # of factors is (1+1)(1+1)=4. Sufficient.

(2) n and 2^3 are each divisible by the same number of positive integers --> 2^3 has 4 different positive factors (1, 2, 4, and 8) so n has also 4. Sufficient.

Hi Bunuel

Here, are not we only considering positive factors. What about negative factors ?

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.
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Re: If n is an integer, then n is divisible by how many positive  [#permalink]

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Re: If n is an integer, then n is divisible by how many positive &nbs [#permalink] 10 Aug 2018, 15:33
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