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# The number 16,000 has how many positive divisors?

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Senior Manager
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The number 16,000 has how many positive divisors? [#permalink]  26 Nov 2017, 09:18
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Question Stats:

66% (00:18) correct 33% (00:20) wrong based on 6 sessions
The number 16,000 has how many positive divisors?

[Reveal] Spoiler: OA
32

Kudos for correct solution.
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Re: The number 16,000 has how many positive divisors? [#permalink]  30 Nov 2017, 06:00
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Bunuel wrote:
The number 16,000 has how many positive divisors?

[Reveal] Spoiler: OA
32

Here 16000 can be written as = $$2^7 X 5^3$$

Now positive factors is = 8 X 4 = 32 (No. of positive factors of $$x^y X m ^n= (y+1) X (n+1)$$)
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Re: The number 16,000 has how many positive divisors? [#permalink]  08 Dec 2017, 11:11
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Expert's post
Bunuel wrote:
The number 16,000 has how many positive divisors?

[Reveal] Spoiler: OA
32

Kudos for correct solution.

----ASIDE-----------------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
-----ONTO the question--------------------------

16,000 = (2^7)(5^3)
So, the number of positive divisors of 16,000 = (7+1)(3+1)
= (8)(4)
= 32

Cheers,
Brent
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Brent Hanneson – Creator of greenlighttestprep.com

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Re: The number 16,000 has how many positive divisors? [#permalink]  14 May 2018, 18:13
GreenlightTestPrep wrote:
Bunuel wrote:
The number 16,000 has how many positive divisors?

[Reveal] Spoiler: OA
32

Kudos for correct solution.

----ASIDE-----------------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
-----ONTO the question--------------------------

16,000 = (2^7)(5^3)
So, the number of positive divisors of 16,000 = (7+1)(3+1)
= (8)(4)
= 32

Cheers,
Brent

I understand this concept but just for in depth clarity please claarify that the above method will list all the factor including negatives and positives ? so I
divided 32/2 to get 16 positive and 16 negative factors ...please clarify?
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Re: The number 16,000 has how many positive divisors? [#permalink]  15 May 2018, 00:55
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Expert's post
In the context of GRE,we stick to factoring positive integers to sidestep negative factor issue. We deal with it in higher mathematics though!

If the number is positive only condsider positive factors. I doubt you will find negative numbers in the factorization problems.
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Re: The number 16,000 has how many positive divisors? [#permalink]  15 May 2018, 20:34
Hwo do we get to 16,000= (2^7)(5^3) Is it trial and error?
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Re: The number 16,000 has how many positive divisors? [#permalink]  15 May 2018, 23:21
Expert's post
kruttikaaggarwal wrote:
Hwo do we get to 16,000= (2^7)(5^3) Is it trial and error?

Final goal is to reach 1.

We can clearly see that in 16000 we would have many multiples of 2. So one way is to keep dividing by 2 till we can divide no more.

16000/2 = 8000 ...... one factor of 2
8000/2 = 4000 ..........second factor 2
.
.
1000/2= 500 .............. 5th factor factor of 2
.
250/2=125 ................. 7th Factor of 2

Now we need other prime factors so we move on to higher prime number such as 3 and 5.

Division by 3 is not possible so 5

125 can be divided 3 times with 5 to get 1.

Hence $$16000=2^7 \times 5^3$$
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Re: The number 16,000 has how many positive divisors? [#permalink]  16 May 2018, 04:22
1
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Expert's post
kruttikaaggarwal wrote:
Hwo do we get to 16,000= (2^7)(5^3) Is it trial and error?

Here's our video explaining how to find the prime factorization of a number:

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Re: The number 16,000 has how many positive divisors? [#permalink]  16 May 2018, 21:38
GreenlightTestPrep wrote:
kruttikaaggarwal wrote:
Hwo do we get to 16,000= (2^7)(5^3) Is it trial and error?

Here's our video explaining how to find the prime factorization of a number:

Is there a way to factor out a big number like this faster on the test, or do you recommend we just go for the smallest prime numbers?
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Re: The number 16,000 has how many positive divisors? [#permalink]  16 May 2018, 21:55
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Expert's post
Ideally you can break the number into smaller number whose factors you know such as

$$16000 = 16 \times 1000$$ Now 1000 is $$10^3$$ or $$2^3 \times 5^3$$ and 16 is $$2^4$$

So 16000 is $$2^4 \times 2^3 \times 5^3$$.
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Re: The number 16,000 has how many positive divisors?   [#permalink] 16 May 2018, 21:55
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