How many positive factors of 2^4*3^5*10^4 are perfect squares which ar

Question

How many positive factors of  2^4*3^5*10^4  are perfect squares which are greater than 1?

A. 41
B. 42
C. 43
D. 44
E. 45

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chetan2u · Accepted Answer

Get the term in prime factorization form...
 2^4*3^5*10^4=2^4*3^5*2^4*5^4=2^8*3^5*5^4

For a perfect square we have to have even powers of the prime factors..
so..
1) Powers of 2.... 2^0, 2^2, 2^4, 2^6, 2^8 ---- 5 numbers
2) Powers of 3.... 3^0, 3^2, 3^4 ---- 3 numbers
3) Powers of 5.... 5^0, 5^2, 5^4 ---- 3 numbers

ways to get 1 or more of these 5*3*3=45..
BUT , we will also have a way where all are with power 0... 2^0*3^0*5^0=1 , and we are looking for >1..

Hence 45-1=44

D

GMATWhizTeam · Answer

Solution:  
 •  2^4*3^5*10^4  
On prime factorizing above, we get
 •  2^4*3^5*2^4*5^4 = 2^8*3^5*5^4  
For the factor to be a perfect square, the power of factor must be an even number
 • For  2^8 , there are 5 cases ( 2^0, 2^2, 2^4,2^6 , and  2^8 )
• For  3^5 , there are 3 cases ( 3^0, 3^2 , and  3^4 )
• For  5^4 , there are 3 cases ( 5^0 ,  5^2 , and  5^4 )
 o Total number of factors with even power =  5*3*3 = 45 
o But, we need to eliminate the case where the factor is  2^0*3^0*5^0 = 1  because we are asked to find the number of factors greater than one
  Thus, required number =  45 – 1 = 44    
Hence, the correct answer is  Option D.

Kinshook · Answer

How many positive factors of  2^4*3^5*10^4  are perfect squares which are greater than 1?

2^4*3^5*10^4 = 2^8*3^5*5^4

Positive factors of  2^4*3^5*10^4  are perfect squares which are greater than 1 are of the form = 2^(0 or 2 or 4 or 6 or 8)*3^(0 or 2 or 4)*5^(0 or 2 or 4) > 1

5*3*3 - 1 = 45 -1 = 44; Since 2^0*3^0*5^0 = 1 is not possible

IMO D

ScottTargetTestPrep · Answer

2^4 x 3^5 x 10^4 = 2^4 x 3^5 x 2^4 x 5^4 =  2^8 x 3^5 x 5^4

Recall that perfect squares have integer bases and even exponents. Thus, there are 5 perfect squares from powers of 2 (2^0, 2^2, 2^4, 2^6 and 2^8), 3 from powers of 3 (3^0, 3^2 and 3^4) and 3 from powers of 5 (5^0, 5^2 and 5^4), so there are a total of 5 x 3 x 3 = 45 perfect square factors in 2^4 x 3^5 x 10^4. However, this also includes the perfect square factor of 1. Therefore, excluding 1, there are 44 perfect square factors in 2^4 x 3^5 x 10^4.

Answer: D

User_user · Answer

sorry what os 2^0 why we take this? Is this square?

Bunuel · Answer

... are perfect squares  which are greater than 1 ?

2^0 = 1, which IS a perfect square but it's not greater than 1 as required per the stem.

vv65 · Answer

Let X = 2^4 * 3^5 * 10^4 = 2^8 * 3^5 * 5^4

See this list: 2, 4, 8, 16 and 3, 9 and 5, 25 
The squares of all these numbers are factors of X
 So we start with 8 perfect square factors greater than 1

From the above list, take a multiple of 2 and a multiple of 3. 
There are 4*2=8 such pairs 
The squares of their products are factors of X
Similarly, we get 8 more pairs with multiples of 2 and 5, and 4 more pairs with multiples of 3 and 5
 That's another 8+8+4=20 perfect square factors greater than 1

From the list, take one multiple of 2, one of 3, and one of 5.
The square of the product of these 3 numbers is a factor of X
There are 4*2*2=16 such triplets 
 So that's 16 more perfect square factors greater than 1

Finally,
 the total number of factors greater than 1 is 8+20+16=44

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How many positive factors of 2^43^510^4 are perfect squares which ar

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