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# How many distinct positive factors does 30,030 have?

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How many distinct positive factors does 30,030 have? [#permalink]

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18 Dec 2012, 04:07
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How many distinct positive factors does 30,030 have?

A. 16
B. 32
C. 64
D. 128
E. 256
[Reveal] Spoiler: OA

Last edited by Bunuel on 18 Dec 2012, 04:26, edited 1 time in total.
Moved to PS forum and added OA.
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Re: How many distinct positive factors does 30,030 have? [#permalink]

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18 Dec 2012, 04:30
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Expert's post
Drik wrote:
How many distinct positive factors does 30,030 have?

A. 16
B. 32
C. 64
D. 128
E. 256

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

BACK OT THE ORIGINAL QUESTION:

Factorize 30,030=2*3*5*7*11*13, thus the number of factors of 30,030 is (1+1)(1+1)(1+1)(1+1)(1+1)(1+1)=2^6=64.

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Re: How many distinct positive factors does 30,030 have? [#permalink]

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08 Feb 2015, 19:27
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How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 13:38
Bunuel wrote:

Factorize 30,030=2*3*5*7*11*13, thus the number of factors of 30,030 is (1+1)(1+1)(1+1)(1+1)(1+1)(1+1)=2^6=64.

How does one do this aspect quickly?
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Re: How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 15:12
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redfield wrote:
Bunuel wrote:

Factorize 30,030=2*3*5*7*11*13, thus the number of factors of 30,030 is (1+1)(1+1)(1+1)(1+1)(1+1)(1+1)=2^6=64.

How does one do this aspect quickly?

Here's a free video lesson on finding the prime factorization of a number: http://www.gmatprepnow.com/module/gmat- ... /video/825
Here's a free video lesson that explains why Bunuel's formula works: http://www.gmatprepnow.com/module/gmat- ... /video/828

Cheers,
Brent
_________________

Brent Hanneson - Founder of GMAT Prep Now, a free & comprehensive GMAT course with:
- over 500 videos (35 hours of instruction)
- over 800 practice questions
- 2 full-length practice tests and other bonus offers
- http://www.gmatprepnow.com/

Manager
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Re: How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 18:47
GMATPrepNow wrote:

Here's a free video lesson on finding the prime factorization of a number: http://www.gmatprepnow.com/module/gmat- ... /video/825
Here's a free video lesson that explains why Bunuel's formula works: http://www.gmatprepnow.com/module/gmat- ... /video/828

Cheers,
Brent

I appreciate the videos which were informative however they don't really answer my specific question; I'm not asking about how to find the # of divisors, I'm wondering how (and this wasn't explained in either video) you quickly figure our the prime factors of a massive number like 30,030?

In the video the question is 14,000 and he just skips to "and here are the prime factors" and I don't get how you figure that out in a timely manner.
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How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 19:17
redfield wrote:
GMATPrepNow wrote:

Here's a free video lesson on finding the prime factorization of a number: http://www.gmatprepnow.com/module/gmat- ... /video/825
Here's a free video lesson that explains why Bunuel's formula works: http://www.gmatprepnow.com/module/gmat- ... /video/828

Cheers,
Brent

I appreciate the videos which were informative however they don't really answer my specific question; I'm not asking about how to find the # of divisors, I'm wondering how (and this wasn't explained in either video) you quickly figure our the prime factors of a massive number like 30,030?

In the video the question is 14,000 and he just skips to "and here are the prime factors" and I don't get how you figure that out in a timely manner.

At 1:30 in the video http://www.gmatprepnow.com/module/gmat- ... /video/825, we explain the process using a tree diagram. The process works for ANY number.

Cheers
Brent
_________________

Brent Hanneson - Founder of GMAT Prep Now, a free & comprehensive GMAT course with:
- over 500 videos (35 hours of instruction)
- over 800 practice questions
- 2 full-length practice tests and other bonus offers
- http://www.gmatprepnow.com/

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Re: How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 19:26
GMATPrepNow wrote:
At 1:30 in the video http://www.gmatprepnow.com/module/gmat- ... /video/825, we explain the process using a tree diagram. The process works for ANY number.

Cheers
Brent

So you see 14,000 and have to do a factor tree starting with a number you can eyeball like 140 and 100 then continue breaking those numbers down?

I'm sorry if I'm missing something here (feel like I'm definitely overcomplicating or simply not getting a simple idea); but when I see a number like 30,030 and one of the steps is "30,030 = 2*3*5*7*11*13" it seems like I'm missing an entire part of the explanation because it seems the speed people are getting these primes would be something more streamlined than a factor tree. It's possible it's just a matter of practice makes it faster I just wasn't sure if I was missing an entire step.

Thank you for the explanations.
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Re: How many distinct positive factors does 30,030 have? [#permalink]

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19 Nov 2015, 21:00
2
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redfield wrote:
GMATPrepNow wrote:
At 1:30 in the video http://www.gmatprepnow.com/module/gmat- ... /video/825, we explain the process using a tree diagram. The process works for ANY number.

Cheers
Brent

So you see 14,000 and have to do a factor tree starting with a number you can eyeball like 140 and 100 then continue breaking those numbers down?

I'm sorry if I'm missing something here (feel like I'm definitely overcomplicating or simply not getting a simple idea); but when I see a number like 30,030 and one of the steps is "30,030 = 2*3*5*7*11*13" it seems like I'm missing an entire part of the explanation because it seems the speed people are getting these primes would be something more streamlined than a factor tree. It's possible it's just a matter of practice makes it faster I just wasn't sure if I was missing an entire step.

Thank you for the explanations.

I can see this is divisible by 10.
So, 30,030 = (3003)(10)
Or 30,030 = (3003)(2)(5)
Well, the sum of the digits is 6, and 6 is divisible by 3, which means 3003 is divisible by 3 (this in an important divisibility rule that's discussed in this free video: http://www.gmatprepnow.com/module/gmat- ... /video/822 )
So, 30,030 = (3)(1001)(2)(5)
This is where it gets a bit tricky since it's hard to see any PRIME divisors of 1001. We know that 2, 3 and 5 don't work. What about 7?
When we check we get: 1001 = (7)(143)

So, 30,030 = (3)(7)(143)(2)(5)
Finally, 143 = ...
So, 30,030 = (3)(7)(11)(13)(2)(5)

Cheers,
Brent
_________________

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- over 500 videos (35 hours of instruction)
- over 800 practice questions
- 2 full-length practice tests and other bonus offers
- http://www.gmatprepnow.com/

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How many distinct positive factors does 30,030 have? [#permalink]

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20 Nov 2015, 08:40
GMATPrepNow wrote:

I can see this is divisible by 10.
So, 30,030 = (3003)(10)
Or 30,030 = (3003)(2)(5)
Well, the sum of the digits is 6, and 6 is divisible by 3, which means 3003 is divisible by 3 (this in an important divisibility rule that's discussed in this free video: http://www.gmatprepnow.com/module/gmat- ... /video/822 )
So, 30,030 = (3)(1001)(2)(5)
This is where it gets a bit tricky since it's hard to see any PRIME divisors of 1001. We know that 2, 3 and 5 don't work. What about 7?
When we check we get: 1001 = (7)(143)

So, 30,030 = (3)(7)(143)(2)(5)
Finally, 143 = ...
So, 30,030 = (3)(7)(11)(13)(2)(5)

Cheers,
Brent

Thank you very much for breaking it down like this, it was a simple matter of the task appearing more daunting to me than it actually was so this step-by-step was perfect thank you Brent.
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Re: How many distinct positive factors does 30,030 have? [#permalink]

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16 Mar 2016, 01:15
Here 1001 is divisible by 11
thats the only basic problem to be solved actually
and also the number of +ve divisors = product of powers of primes after increase them by 1
Re: How many distinct positive factors does 30,030 have?   [#permalink] 16 Mar 2016, 01:15
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