How many factors of 3600 are even?

Question

How many factors of 3600 are even? A) 9 B) 12 C) 24 D) 36 E) 45 Source: https://www.GMATinsight.com

chetan2u · Accepted Answer

3600=2^4*3^2*5^2 ...

Best way is to find ODD factors and then subtract from total..
Total =  (4+1)(2+1)(2+1)=5*3*3=45 
Odd factors =  (2+1)(2+1)=3*3=9

So rest are even factors = 45-9=36

D

Various forms of this question are available

Salsanousi · Answer

3600 = 100 * 36 = 4 * 25 * 6 * 6 = 2^2 * 5^2 * 2^2 * 3^2 = 2^4 * 3^2 * 5^2

Find the total factors - odd factors = even factors

To find factors of a number we find the prime numbers = a^p * b^q...

The number of factors will be = (p+1) * (q+1)...

To find the number of odd factors we take all the other prime factors other than 2 in a similar fashion to the above

We take the powers of odd primes = (2+1)*(2+1) = 9

Total = (4 + 1) (2+1) (2+1) = 5 * 3 * 3 = 45

45 - 9 = 36

Answer choice D

I hope this helps!

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GMATinsight · Answer

Prime factorise the number 3600 i.e.  3600 = 2^4*3^2*5^2 
Row 1:  2^0 ,  2^1, 2^2, 2^3, 2^4  are factors of the number (except  2^0  because for number to be even every factor must have a minimum 21)
Row 2:  3^0, 3^1, 3^2  are factors of the number
Row 3:  5^0, 5^1, 5^2  are factors of the number

Total number of factors = (4)*(3)*(3) = 36 Factors
Answer: Option D

EncounterGMAT · Answer

First make prime factorization of 3600=2^4 * 3^2 * 5^2 where 2, 3, and 5 are prime factors of 4, 2 and 2 are their powers.

The number of factors of 3600 will be expressed by the formula (p+1)(q+1)(r+1) where p,q and r are the powers.

According to the above the number of factors is (4+1)(2+1)(2+1)=45 factors...............................(1)

Now, get rid of powers of 2 as they give even factors, so the remaining left out numbers would be (2+1)(2+1)=9 factors......................(2)

All the remaining factors will be odd, therefore 3600 has 45-9=36 even factors.

Hence, the answer is D.

Hope it helps.

if you like.

ShankSouljaBoi · Answer

Hi  GMATinsight 
Why cant we count normally ? Like 2^0(discarded)
2^1* any factor of 3/5 =1*6
2^2* any factor of 3/5 =1*6
2^3* any factor of 3/5 =1*6
2^4* any factor of 3/5 =1*6

total 24

GMATinsight · Answer

ShankSouljaBoi 
What you are counting as 6 is actually 9 (Combination of all powers of 3 with all powers of 5)

2^1* any factor of 3/5 =1*9
2^2* any factor of 3/5 =1*9
2^3* any factor of 3/5 =1*9
2^4* any factor of 3/5 =1*9

Total = 36

I hope this helps!!!

yashna36 · Answer

Hi,

this was very helpful. Had a doubt though..so is there any way to directly figure the no of even factors? my question being why did we find the odd factors first and then substract from the total instead of directly finding the even?

ShankSouljaBoi · Answer

Please scroll above. I have illustrated the same  yashna36

Posted from my mobile device

GMATinsight · Answer

Your doubt is valid and one solution is posted above but think from this perspective

Who would think that calculating even factors will be more difficult than calculating even factors???

because a layman idea would be to consider calculation of both equally easy/tough therefore it's always good to know both methods.

I hope this helps!!!

GMATinsight · Answer

Alternative:

3600 = 2*1800

i.e. every factor of 1800 when multiplied with 2 will become even factor of 3600

1800 = 2^3*3^2*5^2

factors of  1800 = (3+1)*(2+1)*(2+1) = 4*3*3 = 36

Answer: Option D

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How many factors of 3600 are even?

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How many factors of 3600 are even?

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