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# If the integer n has exactly three positive divisors, including 1 and

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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
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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, including 1 and [#permalink]
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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, including 1 and [#permalink]
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, including 1 and [#permalink]
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
It is 5. (b). If you try and experiment it, any number with three factors total, you will get 4 as one possibility as it is the only possibility there is. so you know that integer n=4. So 2 to the power of 4 is 16. 16 has 5 factors including itself and 1!
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
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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

because n will always be the square of a prime,
positive divisors of n^2 will always be 1, √n, n, n√n, n^2
5
B
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
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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

Since n has exactly 3 positive divisors we can conclude that n is a perfect square of a prime number. For instance, let’s consider the prime number 3. Notice that 3^2 = 9, and the factors of 9 are 1, 3, and 9.

Thus, if we let n = 9, then 9^2 = 81.

The factors of 81 are 1, 81, 9, 27, and 3. Thus, n^2 has 5 positive divisors.

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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
Hi Scott,

Thanks for the explanation but I am bit lost here. In your explaination n^2 has three factors i.e 9 ( 1,3,9) but the question says n has three factors i.e 3 but three has only two factors ( 1,3). Where am I going wrong? Can you also explain the same with one more example?
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
santro789 wrote:
Hi Scott,

Thanks for the explanation but I am bit lost here. In your explaination n^2 has three factors i.e 9 ( 1,3,9) but the question says n has three factors i.e 3 but three has only two factors ( 1,3). Where am I going wrong? Can you also explain the same with one more example?

You should read a question and the solutions MUCH more carefully.

In Scott's example, n = 9 NOT n^2.

n = 9 has three factors: 1, 3, and 9. n^2 in this case will be n^2 = 9^2 = 3^4 and it will have 4 + 1 = 5 factors.
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
I'll explain in a couple of sentences.
If a number has only 3 positive divisors, it means that it is a square of a prime number eg 4,9, 25 etc.
Q) how many factors will n2 have: substitute n for 9.
9^2=81
LCM gives us {1,3,9,27,81} ie 5.

Kudos if helped. I need to reach 25 kudos

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.
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
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Number of factors

If we have to find the number of factors of any number say N, then we should follow below steps:

Step 1: Prime factorize, N = p^a * q^b * r^c...

Step 2: The number of factors of N= (a+1)(b+1)(c+1)…

Now since the integer n has exactly three positive divisors
it means the number of factors = (2+1)
hence n is square of a prime number.

n = p^2
n^2 = p^4

hence, the Number of factors = (4+1) = 5

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.
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]

Solution

Given
• Integer n has exactly three positive divisors, including 1 and n

To find
• The number of positive divisors of n^2

Approach and Working out

Square of a prime number has 3 factors.
• So, n= $$p^2$$
• $$n^2$$= $$p^4$$
o Total factors of $$n^2$$ = 5

Hence, option B is the correct answer.

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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
Let n = 4 [Divisors: 3=> 1 , 2, 4] => $$n^2: 4^2$$ = 16 [Divisors: 5=> 1 , 2, 4, 8, 16]

Let n = 9 [Divisors: 3=> 1 ,3, 9] => $$n^2: 9^2$$ = 81 [Divisors: 5=> 1 , 3, 9, 27, 81]

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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
Lets say n = 4.

4 has three factors: 1, 2, 4

4^2 = 1, 2, 4, 8, 16

4^2 has five factors. Answer is B.
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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
Asked: If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

n is of the form p^2 where p is a prime number
n^2 will be of the form p^4 and will have 5 positive divisors.

IMO B

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Re: If the integer n has exactly three positive divisors, including 1 and [#permalink]
25 works
25^2=625=5^4
4+1=5
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If the integer n has exactly three positive divisors, including 1 and [#permalink]
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

­n can be  4 if you think a bit as it has divisor also known as factor as 1 2 and 4
so n^2 is 16 which is 2^4
hence it has (4+1) factors which is 5
Give kudos if you like this solution
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If the integer n has exactly three positive divisors, including 1 and [#permalink]
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