How many factors of 36^2?

[maxim730]

How many factors does 36^2 have?

A)2
B)8
C)24
D)25
E)26

Answer in a bit.

My solution:

factors of 36 = 2^2 * 3 ^2 = 9 factors when you write it out:
1,2,3,4,,9,12,18,6, and 36

Therefore, 36^2 = 9 * 9 factors = 81 factors.. but 81 isn't a choice..

Thanks!!!

[Vasudha]

See attached..

[BTGmoderatorRO]

let us take the our N=36
N^2=36^2=1296
finding the prime factors of 1296 gives 4*4*9*9
=(2^2) * (2^2) *(3^2) *(3^2)
solving using indices
=(2^4) * (3^4)
using the factor formula f= (p+1)(r+1)
where p and r are the power factors of the prime number
f= (4+1) * (4+1)
=5 * 5 =25

option D is the correct option

[Brent@GMATPrepNow]

How many factors does 36^2 have?
A)2
B)8
C)24
D)25
E)26

-------ASIDE-------------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
---NOW ONTO THE QUESTION-----------------

36 = (2)(2)(3)(3)
So, 36² = (36)(36)
= (2)(2)(3)(3)(2)(2)(3)(3)
= (2^4)(3^4)
So, the number of positive divisors of 36² = (4+1)(4+1)
= (5)(5)
= 25

Answer: D

Cheers,
Brent

[ceilidh.erickson]

let us take the our N=36
N^2=36^2=1296
finding the prime factors of 1296 gives 4*4*9*9
=(2^2) * (2^2) *(3^2) *(3^2)
solving using indices
=(2^4) * (3^4)
using the factor formula f= (p+1)(r+1)
where p and r are the power factors of the prime number
f= (4+1) * (4+1)
=5 * 5 =25

option D is the correct option

You will not have a calculator on the GMAT Quant section. As such, it would be pointless to multiple out 36*36 to get 1296, and then factor it again. This would be a serious waste of time.

As a general rule when you don't have a calculator, you should never make numbers bigger. Always make them smaller!

Break 36*36 down into (4*9)(4*9) and then 2*2*2*2*3*3*3*3 without taking the intermediate step of multiplying it into a larger number.

[ceilidh.erickson]

There is also a shortcut to this particular 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
primes will always have exactly 2 distinct factors - even number

18:
1 x 18
2 x 9
3 x 6
6 distinct factors - even number

36:
1 x 36
2 x 18
3 x 12
4 x 9
6 x 6
9 distinct factors - odd number

Since we're only looking for DISTINCT factors, we only want to count one of the 6's for 36. 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.

[Admin1]

36^2 can be written as a product of prime factors as,
(2 x 2 x 3 x 3)^2
= 2^2 x 2^2 x 3^2 x 3^2
= 2^(2+2) x 3^(2+2)
= 2^4 x 3^4

To find the number of factors, add 1 to each exponent and multiply.
Number of factors = (4 + 1)(4 + 1)
Number of factors = (5)(5) = 25

[Admin1]

Solution:
36^2 = (6^2)^2 = 6^4 = (2 x 3)^4 = (2^4)(3^4)
So, if n is a factor of 36^2, then
n = (2^a)(3^b),
where
0 <= a <= 4, 0 <= b <= 4.
For a, there are five choices: 0, 1, 2, 3, 4. Similarly there are five choices for b. For each choice of a, b, there is a different factor for 36^2.
Thus, there are
5 x 5 = 25 factors in 36^2.

Answer: D

[Scott@TargetTestPrep]

How many factors does 36^2 have?

A)2
B)8
C)24
D)25
E)26

Answer in a bit.

My solution:

factors of 36 = 2^2 * 3 ^2 = 9 factors when you write it out:
1,2,3,4,,9,12,18,6, and 36

Therefore, 36^2 = 9 * 9 factors = 81 factors.. but 81 isn't a choice..

Thanks!!!

Solution:

Since 36^2 = (9 x 4)^2 = (3^2 x 2^2)^2 = 3^4 x 2^4, then 36^2 has (4 + 1)(4 + 1) = 5 x 5 = 25 factors.

Alternate Solution:

We see that 36^2 is a perfect square and recall that a perfect square always has an odd number of factors. Therefore, the only answer choice that can be correct is D.

Answer: D