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Intern  B
Joined: 16 Apr 2009
Posts: 14
How many distinct positive factors does 30,030 have?  [#permalink]

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10 00:00

Difficulty:   65% (hard)

Question Stats: 63% (02:05) correct 37% (01:54) wrong based on 291 sessions

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How many distinct positive factors does 30,030 have?

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

Originally posted by Drik on 18 Dec 2012, 03:07.
Last edited by Bunuel on 18 Dec 2012, 03:26, edited 1 time in total.
Moved to PS forum and added OA.
Math Expert V
Joined: 02 Sep 2009
Posts: 61221
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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1
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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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Current Student S
Joined: 18 Aug 2014
Posts: 323
How many distinct positive factors does 30,030 have?  [#permalink]

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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?
GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4318
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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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
_________________
Current Student S
Joined: 18 Aug 2014
Posts: 323
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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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.
GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4318
How many distinct positive factors does 30,030 have?  [#permalink]

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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
_________________
Current Student S
Joined: 18 Aug 2014
Posts: 323
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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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.
GMAT Club Legend  V
Joined: 11 Sep 2015
Posts: 4318
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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2
1
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
_________________
Current Student S
Joined: 18 Aug 2014
Posts: 323
How many distinct positive factors does 30,030 have?  [#permalink]

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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.
Current Student D
Joined: 12 Aug 2015
Posts: 2537
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: How many distinct positive factors does 30,030 have?  [#permalink]

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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
_________________
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Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 515
Location: India
Re: How many distinct positive factors does 30,030 have?  [#permalink]

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30030 = 2 x 3 x 5 x 7 x 11 x 13

total number of factors = (1+1) (1+1) (1+1) (1+1) (1+1) (1+1) = 2^6 = 64

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

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