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Manager  Joined: 22 Jul 2009
Posts: 131
Location: Manchester UK
How many different factors does the integer n have?  [#permalink]

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1
7 00:00

Difficulty:   15% (low)

Question Stats: 70% (00:55) correct 30% (01:19) wrong based on 202 sessions

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How many different factors does the integer n have?

(1) n = (a^4)(b^3) where a and b are different positive prime numbers.
(2) The only positive prime numbers that are factors of n are 5 and 7.

Originally posted by sagarsabnis on 05 Jan 2010, 12:18.
Last edited by Bunuel on 27 Jul 2015, 08:44, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
##### Most Helpful Expert Reply
Math Expert V
Joined: 02 Sep 2009
Posts: 57022
Re: How many different factors does the integer n have?  [#permalink]

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3
3
shalva wrote:
IMO it's (A)

From Statement 1:

n = a * a * a * a * b * b * b

n has 13 factors: 12 different combinations of a & b + 1.

Statement 2 tells nothing: n could have only 2 prime factors but what about non-prime factors?! we should consider them too. f.e. 35, 25, 49 and so on.

The answer is (A), but n has (4+1)(3+1)=20 factors, including 1 and n itself, not 13.

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.

In original question $$n =a^4*b^3$$, so number of factors =(4+1)(3+1)=20.
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Tuck School Moderator Joined: 20 Aug 2009
Posts: 266
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)
Re: How many different factors does the integer n have?  [#permalink]

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IMO it's (A)

From Statement 1:

n = a * a * a * a * b * b * b

n has 13 factors: 12 different combinations of a & b + 1.

Statement 2 tells nothing: n could have only 2 prime factors but what about non-prime factors?! we should consider them too. f.e. 35, 25, 49 and so on.
Manager  Joined: 22 Jul 2009
Posts: 131
Location: Manchester UK
Re: How many different factors does the integer n have?  [#permalink]

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ohhh I didnt read the question correctly...anyways thanks a lot...
Tuck School Moderator Joined: 20 Aug 2009
Posts: 266
Location: Tbilisi, Georgia
Schools: Stanford (in), Tuck (WL), Wharton (ding), Cornell (in)
Re: How many different factors does the integer n have?  [#permalink]

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Thanks for clarification, I've missed $$a^0$$ and $$b^0$$
Current Student D
Joined: 12 Aug 2015
Posts: 2604
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: How many different factors does the integer n have?  [#permalink]

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1
Nice Question.
Here is what i did in this one.
We need to get the number of factors of positive integer n.

Statement 1=>
As a and b are "different" prime numbers => Number of factors of a must be 5*4=20
Hence sufficient .
Statement 2=>
There exist ∞ numbers with the same set of prime numbers.
E.g
5*7=> Four factors.
5^2*7^2=> Nine factors.
Etc.
Hence not sufficient.

Hence A.

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Intern  B
Joined: 18 Jan 2017
Posts: 35
Re: How many different factors does the integer n have?  [#permalink]

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(1) says that n = (a^4)(b^3), where a and b are different positive prime numbers.

So, with this, clearly we can find the actual value of n, and hence, the actual factors of n.
Non-Human User Joined: 09 Sep 2013
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Re: How many different factors does the integer n have?  [#permalink]

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_________________ Re: How many different factors does the integer n have?   [#permalink] 09 Aug 2019, 19:04
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