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

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

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New post 24 Dec 2013, 02:48
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Re: If n is an integer, then n is divisible by how many positive integers?  [#permalink]

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New post 24 Dec 2013, 02:49
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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.

Answer: D.
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Re: If n is an integer, then n is divisible by how many positive integers?  [#permalink]

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New post 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.

Hence the answer is D.
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Re: If n is an integer, then n is divisible by how many positive integers?  [#permalink]

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New post 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 integers?  [#permalink]

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New post 03 Jun 2014, 07:43
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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 integers?  [#permalink]

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New post 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 divisible by how many positive  [#permalink]

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Re: If n is an integer, then n divisible by how many positive   [#permalink] 12 Jan 2020, 12:07

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

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