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

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

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).

Answer: B.

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.

Answer: B.

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.
_________________

APPROACH #1: Start testing values in order to locate one number that has exactly 3 positive divisors
1 has 1 positive divisor (1). No good.
2 has 2 positive divisors (1 and 2) No good.
3 has 2 positive divisors (1 and 3). No good.
4 has 3 positive divisors (1, 2 and 4). GREAT!

So it could be possible that n = 4

How many positive divisors does n² have?
n² = 4² = 16
The positive divisors of 16 are as follows: 1, 2, 4, 8, 16 (5 divisors)

Answer: B
---------------------------------------

APPROACH #2: Save a tiny bit of time by recognizing that n must be the square of an integer.
Useful properties:
If K is the square of an integer, then K will have an ODD number of positive divisors.
If K is NOT the square of an integer, then K will have an EVEN number of positive divisors.

We're told that n has exactly 3 positive divisors.
Since 3 is ODD, we know that n must be the square of an integer.
So, n COULD be 1 or 4 or 9 or 16 or 25 or.....

At this point, we need only check possible values of n that are squares of integers.
This, however, does not mean we can choose any square have an integer for the value of n. We still need to make sure that n has exactly 3 positive integers.
1 has 1 positive divisor (1). No good.
4 has 3 positive divisors (1, 2 and 4). GREAT!
From this point, our solution is exactly the same as the one covered in APPROACH #1.

Answer: B

Cheers,
Brent

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.

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

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

Bunuel Zindabaad!!!!

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!

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

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.

Answer: B
_________________

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.
_________________

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

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

Answer C




_________________

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.

Correct Answer: Option B
_________________

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]

Answer B

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.
_________________

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

Posted from my mobile device

25 works
25^2=625=5^4
4+1=5

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________