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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] New post 11 Jun 2008, 22:47
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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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-is-an-integer-then-n-is-divisible-by-how-many-positive-164964.html

[Reveal] Spoiler:
OA: D

ii) i understand

but if i)

let prime numbers be 3,4, then there are 4 possible divisors (1,3,4,12)
but if the prime numbers are 1,2, then there are only 2 possible divisors (1,2)

anyone?


nm just realized that 1 is not a prime number.
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Feb 2014, 01:05, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: DS Q [#permalink] New post 11 Jun 2008, 23:02
Expert's post
zetaexmachina wrote:
let prime numbers be 3,4, then there are 4 possible divisors (1,3,4,12)
but if the prime numbers are 1,2, then there are only 2 possible divisors (1,2)

anyone?


nm just realized that 1 is not a prime number.


4 and 1 are not prime numbers.

3,5 - (1,3,5,15)
5,7 - (1,5,7,35)

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Re: If n is an integer, n is divisible by how many positive [#permalink] New post 07 Feb 2014, 17:37
If n is an integer, n is divisible by how many positive integers? -> # factors of n????

i) n is the product of two prime numbers -> n = p^1 * q^1, where p and q are prime; number of factors of n = 4.
ii) n and 2^3 are each divisible by the same number of positive integers -> 2^3 has 4 factors => n has 4 factors.

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Re: If n is an integer, then n is divisible by how many positive [#permalink] New post 08 Feb 2014, 01:05
Expert's post
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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-is-an-integer-then-n-is-divisible-by-how-many-positive-164964.html

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Re: If n is an integer, then n is divisible by how many positive   [#permalink] 08 Feb 2014, 01:05
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