Factors

[prernamalhotra]

Q) How many factors does 36^2 have?

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

Thank you,
Prerna

[GMATinsight]

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]

[GMATinsight]

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

[GMATinsight]

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

[GMATGuruNY]

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

[feedrom]

Amazing! Thanks brilliant Mitch!

[ceilidh.erickson]

Here is an even easier solution to this problem:

When we're looking for DISTINCT FACTORS of an integer, all positive integers have an EVEN number of distinct factors... except for perfect squares! This is because we can count distinct factors in pairs:

5:
1 x 5

18:
1 x 18
2 x 9
3 x 6

36:
1 x 36
2 x 18
3 x 12
4 x 9
6 x 6

Since we're only looking for DISTINCT factors, we only want to count one of the 6's. Perfect squares will always have an ODD number of distinct factors.

In the given question, we know that 36^2 must also have an odd number of distinct factors. Since D is the only odd answer choice, that must be the answer.