Question: If 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

(The answer is b)

I tried to solve this by taking n^2 = n*n

I included {1, n, x} in my "factor box" for n

I just assigned x as an integer to represent the third of the positive divisors of n based on the question statement.

I tried to create a combined factor box for n*n which included 1, n, x, n^2.

I was not really sure how else to approach this example. The solution bring in a root(n) and this is where I would like some clarification. How des that root(n) come in? And the solution explains that in fact root(n) is also an integer.

Also a rule is presented: For any integer to have exactly 3 positive divisors it must be the perfect square of a prime number.

I am hoping that someone can just explain this rule to me - I am not clear on where it comes from and how this relates to the root(n) in the solution.

Thanks,
Carla

Hi Clara,

If I am not wrong you were Manhattan GMAT for the 750 Math quest?

Any Perfect square number will always have Odd number of Factors.

Ok coming to your question.... the number has exactly three divisiors 1, x, n.

Now think when you have three divisors only what does that mean x * x= n

Let's take few examples:

6=2*3 rt...so the factors are 1, 2, 3,6.
8=2*4=1, 2, 4, 8

Similarly we should be able to write n isit's factor form. The questions says that n has exactly 3 divisors...so now let's try to factor n

n =1*n
n=x*??? (what) we don't have any other factor here.--Just read your question about the rule For any integer to have exactly 3 positive divisors it must be the perfect square of a prime number.---I hope the examples answered your question.

that means n=x*x rt? ---from here you can either apporach the problem by using some values such as 4, 9, 25 etc or take the "conceptual method" --I would have picked up the values and found the answer.

Anyway the other method is:
The question asks us for factors of N^2

that mean the factors of n must be squared too...in other words we should be abl

n^2= (1*n)^2 = 1*n^2
or
n^2= (x*x)^2 =x^2 * x^2

So n^2 has factors of 1, x, x^2, n, n^2(which is same as x^4)

So we see that for n^2 we have 5 factors!!!

I hope that helped

Hi,
Thanks - all of your methods were very helpful and clear. I really appreciate your help!
Carla

I was in the advanced sentence correction and the quest for 750.. did you look at those examples by the way? I did the quest for 750 examples this evening.... going to go over solutions tmw hopefully.

Anytime ..let me knw if I can be of any help!!!

I am done will the quest problems...My GMAT is in next 4 days!! So I am almost done with all the questions.

If I am not wrong you were sitting diagonally to me in the Maths 750 Quest rt?

So when are you taking GMAT?

Saurabh Malpani