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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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24 Dec 2013, 02:48
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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.

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

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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.

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

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