How many positive divisors of 12,500 are the square

Question

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

BrentGMATPrepNow · Accepted Answer

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

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    , 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

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

rohit8865 · Answer

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

LHC8717 · Answer

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

BillyZ · Answer

What is the better approach to resolve it within 2 minutes, the official solution seems length?

KrishnakumarKA1 · Answer

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

no of perfect square divisor = (1+1)(2+1) = 6

budaii · Answer

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?

BrentGMATPrepNow · Answer

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

Kimberly77 · Answer

Great detailed explanation  BrentGMATPrepNow . Question how to ensure not to miss out all listings different perfect squares and is there a short cut of calculating to acheive 6 factors? Thanks Brent

BrentGMATPrepNow · Answer

If you look closely at my list, you'll see I have created the list using a systematic approach. 
this should help ensure you don't miss anything.

Kimberly77 · Answer

I see thanks  BrentGMATPrepNow . To clarify 12,500 = 2x2x5x5x5x5x5 = 2^2 5^2 5^2 5. Therefore only one signle 5 here needed here in order for 12,500 to be perfect square. Wondering why is the exponent of 5^(1+1) = 2 rule doesn't work here? Thanks Brent

BrentGMATPrepNow · Answer

Be careful. 12,500 is NOT a perfect square and, more importantly, we aren't trying to make 12,500 a perfect square.

The question reads: How many  positive divisors  of 12,500 are  squares of integers  (aka perfect squares)? 
So out of the many possible divisors of 12,500, we want to count only those divisors that are perfect squares.

sumit99kr · Answer

To find no.of divisors we need to prime factorize 12500.

12500= 5^5*2^2.
Now we need to find only perfect square divisors. 
So, 5^5*2^2 can be written in terms of perfect squares as below-
 
5*(25^2)*(4^1).
Note 25 and 4 are both perfect squares with powers 2 and 1 respectively. 
So the required answer will be 
(2+1)*(1+1) = 6.

Posted from my mobile device

Kinshook · Answer

Asked: How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

12500 = 2^2*5^5

Sum of Divisors of 12500 which are squares of integers = (1 + 2^2)(1+5^2 + 5^4) 
Number of divisors of 12500 which are squares of integers = 2*3 = 6

IMO C

Kimberly77 · Answer

Ah I see. Knew I was missing something

. Thanks Brent BrentGMATPrepNow for the clarification.

ScottTargetTestPrep · Answer

Let's prime factorize 12,500: 2^2 * 5^5

The largest perfect square factor of 12,500 is 2^2 * 5^4 = 2,500. Thus, if k^2 is a factor of 12,500, then k must be a factor of √2,500 = 50. Since 50 = 2 * 5^2 has (1 + 1)(2 + 1) = 6 factors, 12,500 has 6 factors which are perfect squares.

Answer: C

mysterymanrog · Answer

12,500=125*10^2=5^3*10^2=5^5*2^2*1

Ways to make a perfect square:

1^2, 2^2, 2^4, 5^2, 5^4, 10^2,5^4*2^2.

For a total of 6 divisors.

C

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How many positive divisors of 12,500 are the square

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