OG 11th Edition PS#241-Need explanation

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joyseychow: Master | Next Rank: 500 Posts; Posts: 125; Joined: Mon Dec 15, 2008 9:24 pm

by joyseychow » Thu Apr 02, 2009 7:15 pm

If the integer n has exactly 3 positive divisors, including 1 and n, how many positive divisors does n^2 have?

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

I don't understand why answer is B. Book explanation not clear.
Thanks!

Last edited by joyseychow on Wed Apr 29, 2009 1:36 am, edited 1 time in total.

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Source: Beat The GMAT — Problem Solving | Thu Apr 02, 2009 7:15 pm

Vemuri: Legendary Member; Posts: 682; Joined: Fri Jan 16, 2009 2:40 amFollowed by: 1 member; Thanked: 32 times

Re: OG 11th Edition PS#214-Need explanation

by Vemuri » Thu Apr 02, 2009 8:04 pm

Interesting question. I am not very good with number theories, so I could not come up with a number that has exactly 3 divisors. So, my solution is more theoretical.

Given, n has 1, n & some number 'x' as its divisors. So, n^2 will have 1, n, n^2, x & x^2 as its divisors. So, there are 5 divisors in all.

BTW, please use spoiler when giving answer. This will allow others to give a fair try at the question before providing the explanation.

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vittalgmat: Legendary Member; Posts: 621; Joined: Wed Apr 09, 2008 7:13 pmFollowed by: 4 members; Thanked: 33 times

by vittalgmat » Thu Apr 02, 2009 10:32 pm

Interesting question indeed.

This problem can be easily solved once we know this key concept.

1. A prime number N has ONLY 2 divisors N and 1.
2. Square of a prime number has 3 divisors
ie N^2 has N^2, N and 1 as its divisors.

Now u can easily solve this problem.

let the number be 4. It has the following divisors 4, 2 and 1.

Square of this number is 16. Lets find the divisors of 16 as follows.

1 -------- 16
2 -------- 8
4 -------- 4

So the divisors are 1, 2, 4, 8, 16
So the number of divisors = 5

HT Helps
-V

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joyseychow: Master | Next Rank: 500 Posts; Posts: 125; Joined: Mon Dec 15, 2008 9:24 pm

by joyseychow » Fri Apr 03, 2009 5:54 am

vittalgmat wrote:Interesting question indeed.

This problem can be easily solved once we know this key concept.

1. A prime number N has ONLY 2 divisors N and 1.
2. Square of a prime number has 3 divisors
ie N^2 has N^2, N and 1 as its divisors.

Now u can easily solve this problem.

let the number be 4. It has the following divisors 4, 2 and 1.

Square of this number is 16. Lets find the divisors of 16 as follows.

1 -------- 16
2 -------- 8
4 -------- 4

So the divisors are 1, 2, 4, 8, 16
So the number of divisors = 5

HT Helps
-V

Thanks V for the in depth explanation! Will remember the concept.

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