It is currently 08 Dec 2019, 02:29
My Tests

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

If 78 and 66 are both factors of x, what is the smallest num

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Intern
Intern
Joined: 26 Nov 2019
Posts: 1
Followers: 0

Kudos [?]: 0 [0], given: 0

If 78 and 66 are both factors of x, what is the smallest num [#permalink] New post 26 Nov 2019, 23:25
00:00

Question Stats:

100% (00:38) correct 0% (00:00) wrong based on 1 sessions
If 78 and 66 are both factors of x, what is the smallest number of factors x could have in total?


[Reveal] Spoiler: OA
16

Last edited by Carcass on 28 Nov 2019, 08:14, edited 3 times in total.
Edited by Carcass
Founder
Founder
User avatar
Joined: 18 Apr 2015
Posts: 8962
Followers: 178

Kudos [?]: 2117 [0], given: 8308

CAT Tests
Re: If 78 and 66 are both factors of x, what is the smallest num [#permalink] New post 27 Nov 2019, 02:41
Expert's post
Follow the rules for posting. A numeric entry question must be posted under the right forum NOT in general quant.

The factors are: \(2^2,3,11,39\)

Adding 1 to every exponent and we do have : \(2^3,3 = 8 \times 3=24\)

Total number of factors are 24
_________________

Need Practice? 20 Free GRE Quant Tests available for free with 20 Kudos
GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests.

1 KUDOS received
Intern
Intern
Joined: 04 Oct 2019
Posts: 30
Followers: 0

Kudos [?]: 12 [1] , given: 1

CAT Tests
Re: If 78 and 66 are both factors of x, what is the smallest num [#permalink] New post 28 Nov 2019, 07:27
1
This post received
KUDOS
Carcass wrote:
Follow the rules for posting. A numeric entry question must be posted under the right forum NOT in general quant.

The factors are: \(2^2,3,11,39\)

Adding 1 to every exponent and we do have : \(2^3,3 = 8 \times 3=24\)

Total number of factors are 24



The answer should be 36.

The factors are: 2^2 X 3^2 X 11 X 13

After adding 1 to exponents, the number of divisors = 3 X 3 X 2 X 2 = 36
GRE Instructor
User avatar
Joined: 10 Apr 2015
Posts: 2608
Followers: 95

Kudos [?]: 2813 [0], given: 45

CAT Tests
Re: If 78 and 66 are both factors of x, what is the smallest num [#permalink] New post 28 Nov 2019, 08:11
Expert's post
Kenny1000 wrote:
Can anyone help with this problem?

Its a fill in the box question, there are no options.

If 78 and 66 are both factors of x, what is the smallest number of factors x could have in total?


[Reveal] Spoiler: OA
24


-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:

If k is a factor of N, then k is "hiding" within the prime factorization of N

Consider these examples:
3 is a factor of 24, because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a factor of 70 because 70 = (2)(5)(7)
And 8 is a factor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a factor of 630 because 630 = (2)(3)(3)(5)(7)
-----BACK TO THE QUESTION!---------------------

GIVEN: 78 is a factor of x
78 = (2)(3)(13)
This means 2, 3 and 13 must be in the prime factorization of x
In other words: x = (2)(3)(13)(?)(?)(?)
Please note that the (?)'s represent additional prime numbers that could also be in the prime factorization of x. However, at this point, all we know for certain is that 2, 3 and 13 must be in the prime factorization of x

GIVEN: 66 is a factor of x
66 = (2)(3)(11)
This means 2, 3 and 11 must be in the prime factorization of x

From the earlier information, we already know that x = (2)(3)(13)(?)(?)(?)
Since we already have a 2 and a 3 in the prime factorization of x, we don't need to add more 2's or 3's
But we do need to add 11 to the prime factorization
In other words: x = (2)(3)(13)(11)(?)(?)

So, the smallest possible value of x that meets both conditions is x = (2)(3)(13)(11)
As we can see, 78 is a factor of x because x = (2)(3)(13)(11)
We can also see that 66 is a factor of x because x = (2)(3)(13)(11)

Now that we know the smallest possible value of x, we can apply a nice rule for finding the total number of factors have a positive number.

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

In our case, x = (2^1)(3^1)(13^1)(11^1)
So, the number of positive divisors of x = (1+1)(1+1)(1+1)(1+1) =(2)(2)(2)(2) = 16

Answer: 16 (I have edited the official answer to reflect my solution above)

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com
Sign up for my free GRE Question of the Day emails

Founder
Founder
User avatar
Joined: 18 Apr 2015
Posts: 8962
Followers: 178

Kudos [?]: 2117 [0], given: 8308

CAT Tests
Re: If 78 and 66 are both factors of x, what is the smallest num [#permalink] New post 28 Nov 2019, 08:14
Expert's post
I counted one more exponent but it was elegant :) as a solution.
_________________

Need Practice? 20 Free GRE Quant Tests available for free with 20 Kudos
GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests.

Re: If 78 and 66 are both factors of x, what is the smallest num   [#permalink] 28 Nov 2019, 08:14
Display posts from previous: Sort by

If 78 and 66 are both factors of x, what is the smallest num

  Question banks Downloads My Bookmarks Reviews Important topics  


GRE Prep Club Forum Home| About| Terms and Conditions and Privacy Policy| GRE Prep Club Rules| Contact

Powered by phpBB © phpBB Group

Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®.