First Method:

36^2 = (9*4)^2 = 3^4 * 2^4

Number of factors = (Power of first prime factor + 1) x ((Power of seconf prime factor + 1).....

therefore number of factors = (4+1) x (4+1) = [spoiler]25 Answer[/spoiler]

Second Method:

36^2 = 1296

Which can be broken up into product of two numbers in the following ways

1 x 1296 2 factors
2 x 648 2 factors
3 x 432 2 factors
4 x 324 2 factors
6 x 216 2 factors
8 x 162 2 factors
9 x 144 2 factors
12 x 108 2 factors
16 x 81 2 factors
18 x 72 2 factors
24 x 54 2 factors
27 x 48 2 factors
36 x 36 1 factors

Total factors = 25 ANSWER

For understanding the number of factors formula and it's derivation you can refer to the below mentioned images

prernamalhotra wrote:Q) How many factors does 36^2 have?

1)2
2)8
3)24
4)25
5)26

Thank you,
Prerna

To determine the number of positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply

36^2 = 2^4 * 3^4.
Adding 1 to each exponent and multiplying, we get :
(4+1)(4+1) = 25 factors.

The correct answer is D.

Here's the reasoning.
To determine how many factors can be created from 36^2 = 2^4 * 3^4, we need to determine the number of options for each prime factor:

For 2, we can use 2^0, 2^1, 2^2, 2^3, or 2^4, giving us 5 OPTIONS.
For 3, we can use 3^0, 3^1, 3^2, 3^3, or 3^4, giving us 5 OPTIONS.

Multiplying the number of options for each prime factor, we get:
5*5 = 25 possible factors.

Similar problems:

https://www.beatthegmat.com/divisors-t85731.html
https://www.beatthegmat.com/all-factors- ... 15019.html

A problem about counting only the ODD factors:
https://www.beatthegmat.com/gmat-loves-f ... 72876.html

Amazing! Thanks brilliant Mitch!