How many positive factors of 2^5*3^6*5^2 are perfect squares?

Question

How many positive factors of  2^5*3^6*5^2  are perfect squares?

A. 8
B. 12
C. 18
D. 24
E. 36

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GMATinsight · Accepted Answer

CONCEPT : Perfect squares have Even exponents of all distinct prime factors of the Number

2^5*3^6*5^2 = (2^2)^2*(3^2)^3*(5^2)^1*2

2^2 = a 
 3^2 = b 
 5^2 = c

The factors will be perfect squares if we find the factors of  (a)^2*(b)^3*(c)^1

If  N = a^p*b^q*c^r... 
where a, b, c... are distinct primes
Number of factors of  N = (p+1)*(q+1)*(r+1)*...

i.e. Factors of  (a)^2*(b)^3*(c)^1  will be  (2+1)*(3+1)*(1+1) = 24

Answer: Option D

To know the derivation of Property of ways of finding factors of a number is available in th video here

https://www.youtube.com/watch?v=uecuIrvaXGM

vinitgianchandani · Answer

Make pairs of the squares!

Showing (2^4)*(3^6)*(5^2) as (2^2)*(3^3)*(5^1) (i.e. (2^a)*(3^b)*(5^c))

No. Of factors = (a+1)*(b+1)*(c+1) = (2+1)*(3+1)*(1+1) = 24

GMATWhizTeam · Answer

Solution: 
 •  2^5*3^6*5^2 
 o We need to find the factors of the above number which are perfect square.
o We need to the factors with even power for this. 
•  2^5  has 3 factors with even power i.e.,  2^0, 2^2 , and  2^4 .
•  3^6  has 4 factors with even power i.e.,  3^0, 3^2, 3^4,  and  3^6 
•  5^2  has two factors with even power i.e.,  5^0  and  5^2 
 o Required number =  3*4*2 = 24 .  
Hence, the correct answer is  Option D.

Archit3110 · Answer

pair of square of  2^5*3^6*5^2

(2^2)^2 * (3^2)^3 * (5^2)^1 * 2
factors of squares ; 3*4*2 ; 24
OPTION D

preetamsaha · Answer

How many positive factors of 2^5∗3^6∗5^2 are perfect squares?

Any factor of 2^5∗3^6∗5^2 which is a perfect square will be of the form 2^a∗3^b∗5^c where 
a can be 0 or 2 or 4 (3 ways)
b can be 0 or 2 or 4 or 6 (4 ways)
c can be 0 or 2 (2 ways)
so,required number of factors=3×4×2=24

correct answer D

Scholar94 · Answer

If the formula for finding number of factor does not immediate come to you, another 1 minute approach to solve this is to write out the obvious:

The following are individual perfect squares that are obtainable from each of the component

2^2 
 (2^2)^2 
 (3^3)^2 
 5^2

With this 4 I can make 4! combination of perfect square. 4! is 24. D.

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