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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 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 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 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


What is the better approach to resolve it within 2 minutes, the official solution seems length?
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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
1
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
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How many positive divisors of 12,500 are the square  [#permalink]

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New post 15 Sep 2018, 00:33
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


I don't know if I got a lucky guess with the method I used, but I obtained answer C) in the following manner:

We can find the Prime Factors of 12 500 which are 5^5 and 2^2.

The question asks to obtain the number of positive, perfect square divisors of 12, 500. To do so, we must apply the rule that "The Prime Factorization of a perfect square will have an even number of each prime".

Given this rule, we can see that the PFs of 12 ,500 = (5^5) (2^2). However, it is clear that 5^5 is ODD, which means that the number of positive, perfect square divisors of 12, 500 must be even. Therefore we can infer that the correct answer must be the next best EVEN pair of Prime Factors, that being: (5^4) (2^2).

This gives us a total of 6 numbers if we add the exponents.

Hopefully someone can clarify whether my approach is suitable or perhaps I was just lucky?
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Re: How many positive divisors of 12,500 are the square  [#permalink]

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New post 15 Sep 2018, 06:12
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budaii 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

*kudos for all correct solutions


I don't know if I got a lucky guess with the method I used, but I obtained answer C) in the following manner:

We can find the Prime Factors of 12 500 which are 5^5 and 2^2.

The question asks to obtain the number of positive, perfect square divisors of 12, 500. To do so, we must apply the rule that "The Prime Factorization of a perfect square will have an even number of each prime".

Given this rule, we can see that the PFs of 12 ,500 = (5^5) (2^2). However, it is clear that 5^5 is ODD, which means that the number of positive, perfect square divisors of 12, 500 must be even. Therefore we can infer that the correct answer must be the next best EVEN pair of Prime Factors, that being: (5^4) (2^2).

This gives us a total of 6 numbers if we add the exponents.

Hopefully someone can clarify whether my approach is suitable or perhaps I was just lucky?


Unfortunately, it was only a coincidence that adding the exponents of (5^4)(2^2) yielded the correct answer.

Notice that, if we were to determine how many divisors of (5^4) are squares of integers, we'd conclude (using your approach) that there are 4 such divisors, while there are actually only 3.

Likewise, if we were to determine how many divisors of (2^2)(3^2)(5^2) are squares of integers, we'd conclude (using your approach) that there are only 6 such divisors, while there are actually 8.

Cheers,
Brent
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Re: How many positive divisors of 12,500 are the square &nbs [#permalink] 15 Sep 2018, 06:12
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