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

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Bunuel
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How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   09 Apr 2020, 06:29
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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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D
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   09 Apr 2020, 07:17
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Bunuel wrote:
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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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
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   09 Apr 2020, 07:54
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Bunuel wrote:
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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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.
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How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   10 Apr 2020, 07:30
Bunuel wrote:
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


Are You Up For the Challenge: 700 Level Questions


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
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   13 Apr 2020, 04:44
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Bunuel wrote:
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



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
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   13 Apr 2020, 06:35
sorry what os 2^0 why we take this? Is this square?
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   13 Apr 2020, 06:38
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User_user wrote:
sorry what os 2^0 why we take this? Is this square?


... 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.
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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink] New post   15 Sep 2022, 21:29
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How many positive factors of (2^4 * 3^5 * 10^4) are perfect squares which are greater than 1?

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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Re: How many positive factors of 2^4*3^5*10^4 are perfect squares which ar [#permalink]
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