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How many positive divisors of 12,500 are the square

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How many positive divisors of 12,500 are the square [#permalink]

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New post 13 Feb 2017, 10:27
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How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

*kudos for all correct solutions

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Re: How many positive divisors of 12,500 are the square [#permalink]

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New post 13 Feb 2017, 12:19
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GMATPrepNow wrote:
How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

*kudos for all correct solutions



12500 = 5^5*2^2
thus divisors which are square of integers are
(1)5^2
(2)5^4
(3)2^2
(4)2^2*5^2
(5)2^2*5^4
(6) 1

total 6

Ans C
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Re: How many positive divisors of 12,500 are the square [#permalink]

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New post 13 Feb 2017, 13:05
2
GMATPrepNow wrote:
How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

*kudos for all correct solutions


12500 = 125 * 100 = 5^3 * 2^2 * 5^2 = 5^5 * 2^2

=> We got 5^2, 5^4 and 2^2 as perfect square

=> (2+1)*(1+1) = 6 Answer D
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Re: How many positive divisors of 12,500 are the square [#permalink]

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New post 14 Feb 2017, 06:39
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GMATPrepNow wrote:
How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve



-------------------------------------------------------------------------------

IMPORTANT CONCEPT: The prime factorization of a perfect square will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

--now onto the question-----------------------------------------------------------------------------

12,500 = 2x2x5x5x5x5x5 = (2x2)(5x5)(5x5)(5)
Since we need an even number of each prime [in order for the product to be a perfect square], we need only determine how many different perfect squares can be achieved by using various configurations of (2x2), (5x5) and (5x5)

Let's list them:
1) (2x2) = 4
2) (2x2)(5x5) = 100
3) (2x2)(5x5)(5x5) = 2500
4) (5x5) = 25
5) (5x5)(5x5) = 625
6) 1 [a factor of all positive integers]

So, there are 6 factors of 12,500 that are squares of integers
Answer: C
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Re: How many positive divisors of 12,500 are the square [#permalink]

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New post 20 Mar 2017, 19:40
GMATPrepNow wrote:
GMATPrepNow wrote:
How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve



-------------------------------------------------------------------------------

IMPORTANT CONCEPT: The prime factorization of a perfect square will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

--now onto the question-----------------------------------------------------------------------------

12,500 = 2x2x5x5x5x5x5 = (2x2)(5x5)(5x5)(5)
Since we need an even number of each prime [in order for the product to be a perfect square], we need only determine how many different perfect squares can be achieved by using various configurations of (2x2), (5x5) and (5x5)

Let's list them:
1) (2x2) = 4
2) (2x2)(5x5) = 100
3) (2x2)(5x5)(5x5) = 2500
4) (5x5) = 25
5) (5x5)(5x5) = 625
6) 1 [a factor of all positive integers]

So, there are 6 factors of 12,500 that are squares of integers
Answer: C


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Re: How many positive divisors of 12,500 are the square [#permalink]

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New post 21 Mar 2017, 03:06
GMATPrepNow wrote:
How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

*kudos for all correct solutions


12500 = 2^2 . 5^5 =4^1 . 25^2 .5

no of perfect square divisor = (1+1)(2+1) = 6
Re: How many positive divisors of 12,500 are the square   [#permalink] 21 Mar 2017, 03:06
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How many positive divisors of 12,500 are the square

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