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How many different factors does the integer n have?

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How many different factors does the integer n have?  [#permalink]

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New post Updated on: 27 Jul 2015, 08:44
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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.
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Re: How many different factors does the integer n have?  [#permalink]

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New post 06 Jan 2010, 04:29
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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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Re: How many different factors does the integer n have?  [#permalink]

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New post 05 Jan 2010, 14:01
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.
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Re: How many different factors does the integer n have?  [#permalink]

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New post 05 Jan 2010, 14:37
ohhh I didnt read the question correctly...anyways thanks a lot...
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Re: How many different factors does the integer n have?  [#permalink]

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New post 06 Jan 2010, 04:57
Thanks for clarification, I've missed \(a^0\) and \(b^0\)
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Re: How many different factors does the integer n have?  [#permalink]

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New post 14 Jan 2017, 04:54
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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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Re: How many different factors does the integer n have?  [#permalink]

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New post 19 Jan 2017, 07:30
(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.
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Re: How many different factors does the integer n have?  [#permalink]

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