OG 11th Edition PS#241-Need explanation

[joyseychow]

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!

[Vemuri]

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.

[vittalgmat]

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

[joyseychow]

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.