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# If the integer n has exactly three positive divisors, includ

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If the integer n has exactly three positive divisors, includ [#permalink]  18 Jan 2013, 08:43
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If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

OG 11 #241.

Would someone mind explaining? I'm not satisfied with the explanation in the OG.
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  18 Jan 2013, 09:47
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GMATBeast wrote:
If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

OG 11 #241.

Would someone mind explaining? I'm not satisfied with the explanation in the OG.

Important property: the number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square. (A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square).

Hence, since given that n has 3 (odd) divisors then n is a perfect square, specifically square of a prime. The divisors of $$n$$ are: $$1$$, $$\sqrt{n}=prime$$ and $$n$$ itself. So, $$n$$ can be 4, 9, 25, ... For example divisors of 4 are: 1, 2=prime, and 4 itself.

Now, $$n^2=(\sqrt{n})^4=prime^4$$, so it has 4+1=5 factors (check below for that formula).

Else you can just plug some possible values for $$n$$: say $$n=4$$ then $$n^2=16=2^4$$ --> # of factors of 2^4 is 4+1=5.

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.

So, the # of factors of x=a^2*b^3, where a and b are different prime numbers is (2+1)(3+1)=12.

Hope it's clear.
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  18 Jan 2013, 09:09
Basically, the description says that this is the square of a prime number. So if you square that number, you will have a prime number raised to the fourth power.

That will have 5 factors. For a more detailed description, we have a free factors and multiples lesson on our site.
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  19 Jan 2013, 09:25
quite simple..
take the example of 4...
it has 3 positive divisors (1,2,4)

Now, take the example of 16...
it has only 5 divisors..
so B is the ans
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  30 Jan 2013, 01:02
gmat dose not requires us to remember much.

pick some numbers and see that the number must be a square of prime.

from this departure, we can infer B.

hard one
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  03 Sep 2014, 05:50
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Re: If the integer n has exactly three positive divisors, includ [#permalink]  26 Nov 2015, 03:32
Hello from the GMAT Club BumpBot!

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Re: If the integer n has exactly three positive divisors, includ [#permalink]  01 Dec 2015, 14:42
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Re: If the integer n has exactly three positive divisors, includ   [#permalink] 01 Dec 2015, 14:42
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