How many different perfect squares are factors of 2007^{2007} ?

A) 2016032

B) 2016031

C) 2016030

D) 2016029

E) 2016028

jonathanxu999
Dec 9, 2017

#1**0 **

We can factor the base of 2007 as follows:

2007 =3^2 x 223

Now we can raise the above 2 factors to the power of 2007:

(3^2)^2007 x 223^2007

3^4014 x 223^2007. So from the 4014 exponent, we have 4014/2 =**2007 Perfect Squares**. And from the 2007 exponent, we have: 2007 -1 / 2 =**1,003 Perfect Squares**. Then we multiply them together, we get: 2007 x 1003=**2,013,021 Perfect Squares. **To this will add the sum of the same number of perfect squares, i.e., 2007 + 1003 =**3010 Perfect Squares.**

**So the final total =[1,003 x 2,007] + 3,010 =2,016,031 in total.**

**And that is my attempt !!!.**

**P.S. I neglected to take "1" as a perfect square !!. Therefore, if we add 1 to the above total, we have: 2,016,031 + 1 =2,016,032 !!.**

Guest Dec 9, 2017

edited by
Guest
Dec 9, 2017

edited by Guest Dec 9, 2017

edited by Guest Dec 9, 2017

edited by Guest Dec 9, 2017

edited by Guest Dec 9, 2017

#2**0 **

Look at the problem with this simple example:

2^4 x 3^4=2 "2s" and 2 "3s" =2 x 2 =4 squares. But the product of 2 x 3=6^4, which gives us another 4 squares for a total of 2 x 2 + 4 + 1=9 Perfect squares. So that if we find ALL the divisors of 2^4 x 3^4 =1,296.

ALL divisors of 1,296 =**1 **| 2 | 3** | 4 **| 6 | 8 |** 9** | 12 | **16 **| 18 | 24 | 27 | **36** | 48 | 54 | 72 |** 81** | 108 | **144** | 162 | 216 |** 324** | 432 | 648 | **1296** (25 divisors) =**9 Perfect squares.**

Guest Dec 9, 2017

#3**0 **

Prime factorize 2007 to get 3^2 * 223

This means that there are already 2007 squares in 2007^2007, due to the 3^2.

Divide 2007 by 2 to get the number of perfect squares for the 223, the second part of the prime factorization.

2007/2=1003

After this, you multiply 2007 by 1003, since the prime factorization is 3^2 multiplied by 223.

2007*1003=2013021

Then, you add the original 2007, since it it is 2007^2007, so there are 2007 perfect squares right there already for the 3^2 and add 1003 for the perfect squares from 223.

Add 1 also, because 1 is technically considered to be a perfect square (1*1=1)

2013021+2007+1003+1=2016032

Therefore, there are 2016032 factors in 2007^2007 that are perfect squares

Yahoo answers will always be better!

Guest Dec 10, 2017