GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 24 May 2019, 16:18

### 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

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

# Find the number that divides 16!+1?

Author Message
TAGS:

### Hide Tags

Manager
Joined: 08 Sep 2010
Posts: 163
Location: India
WE 1: 6 Year, Telecom(GSM)
Find the number that divides 16!+1?  [#permalink]

### Show Tags

20 Oct 2010, 03:52
4
40
00:00

Difficulty:

45% (medium)

Question Stats:

61% (01:20) correct 39% (01:45) wrong based on 479 sessions

### HideShow timer Statistics

Find the number that divides 16!+1?

A. 7
B. 11
C. 17
D. 6
E. 18
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: Find the number..the fastest way to solve?  [#permalink]

### Show Tags

20 Oct 2010, 04:02
16
14
ankitranjan wrote:
Find the number that divides 16!+1 or (factorial 16 + 1).

1.7
2.11
3.17
4.6
5.18

Consider KUDOS if u like the question.
I will provide the fastest way if i dont get from anyone.Thanks.

$$16!$$ and $$16!+1$$ are consecutive integers. Consecutive integers are co-prime, which means that they do not share any common factor but 1.

Now, obviously $$16!$$ has 6, 7, 11, and 18 as its factors so $$16!+1$$ won't have any of those. So the only answer choice C (17) is left which might be a factor of $$16!+1$$.

_________________
Manager
Joined: 08 Sep 2010
Posts: 163
Location: India
WE 1: 6 Year, Telecom(GSM)
Re: Find the number..the fastest way to solve?  [#permalink]

### Show Tags

20 Oct 2010, 04:11
10
3
Bunuel wrote:
ankitranjan wrote:
Find the number that divides 16!+1 or (factorial 16 + 1).

1.7
2.11
3.17
4.6
5.18

Consider KUDOS if u like the question.
I will provide the fastest way if i dont get from anyone.Thanks.

$$16!$$ and $$16!+1$$ are consecutive integers. Consecutive integers are co-prime, which means that they do not share any common factor but 1.

Now, obviously $$16!$$ has 6, 7, 11, and 18 as its factors so $$16!+1$$ won't have any of those. So the only answer choice C (17) is left which might be a factor of $$16!+1$$.

Bunuel You are the best.But there is one theorem that is Wilson's theorem...It states that
If n is a prime number ,(n-1)!+1 is divisible by n.

Hence 16!+1 i.e (17-1)! + 1 will be divisible by 17.

Consider Kudos if u find this interesting.
##### General Discussion
Intern
Joined: 09 Aug 2010
Posts: 6
Concentration: Finance
GMAT 1: 660 Q49 V31
Re: Find the number..the fastest way to solve?  [#permalink]

### Show Tags

20 Oct 2010, 04:09
Wow, I was always confused with these question. Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: Find the number..the fastest way to solve?  [#permalink]

### Show Tags

20 Oct 2010, 04:26
2
1
ankitranjan wrote:
Bunuel You are the best.But there is one theorem that is Wilson's theorem...It states that
If n is a prime number ,(n-1)!+1 is divisible by n.

Hence 16!+1 i.e (17-1)! + 1 will be divisible by 17.

Consider Kudos if u find this interesting.

Yes, Wilson's theorem works for this particular number. Though you won't need this theorem for GMAT.
_________________
Current Student
Joined: 12 Aug 2015
Posts: 2617
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

16 Mar 2016, 10:09
1
Using the logic => Multiple +non multiple = non multiple => discarding all the options => only 17 is the viable choose..
Remember => we dont really have to compute the values here.. All is in the logic
_________________
Intern
Joined: 17 Aug 2016
Posts: 48
Find the number that divides 16!+1?  [#permalink]

### Show Tags

04 Oct 2016, 10:08
Bunuel could you advise me if the following is a correct solution or if I am inventing math?

16! = 17!/17 --> 17!/17 + 1 =(17!+17)/17 ---> 16!+1 divisible by 17
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

05 Oct 2016, 02:15
bazu wrote:
Bunuel could you advise me if the following is a correct solution or if I am inventing math?

16! = 17!/17 --> 17!/17 + 1 =(17!+17)/17 ---> 16!+1 divisible by 17

16!+1=(17!+17)/17 but how do you conclude that (17!+17)/17 is divisible by 17? In other words how do you know that [(17!+17)/17]/17 is an integer?
_________________
Intern
Joined: 17 Aug 2016
Posts: 48
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

05 Oct 2016, 09:33
Bunuel wrote:
bazu wrote:
Bunuel could you advise me if the following is a correct solution or if I am inventing math?

16! = 17!/17 --> 17!/17 + 1 =(17!+17)/17 ---> 16!+1 divisible by 17

16!+1=(17!+17)/17 but how do you conclude that (17!+17)/17 is divisible by 17? In other words how do you know that [(17!+17)/17]/17 is an integer?

oh, yes I see, what I had in my mind does't actually make sense! thanks!
Manager
Joined: 12 Oct 2012
Posts: 111
WE: General Management (Other)
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

03 Dec 2016, 07:19
Learnt a new concept.
But when I apply the same to smaller factorials such as 5!+1 = 121 (divisible by 11 and not by 6-because of 3 & 2 in 5! but what about 7).

Is there any exception to the theorem?
Retired Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1353
Location: Viet Nam
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

03 Dec 2016, 10:33
1
Mbawarrior01 wrote:
Learnt a new concept.
But when I apply the same to smaller factorials such as 5!+1 = 121 (divisible by 11 and not by 6-because of 3 & 2 in 5! but what about 7).

Is there any exception to the theorem?

You could see this theorem here: https://en.wikipedia.org/wiki/Wilson's_theorem

The theorem is applied for prime p and (p-1)!+1 divisible by p.

Hence, 4!+1 is divisible by 5. However, 5!+1 will not be divisible by 6.

This theorem could be proved by using advanced mathematic tools, thus this theorem is too hard and we no need to learn this theorem in solving GMAT PS/DS questions.
_________________
Current Student
Joined: 20 Jan 2017
Posts: 58
Location: United States (NY)
Schools: CBS '20 (A)
GMAT 1: 750 Q48 V44
GMAT 2: 610 Q34 V41
GPA: 3.92
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

29 Jan 2017, 18:23
1) Two consecutive integers do not have any factors in common other than 1. This means that factor of 16!+1 has to be a number that is not a factor of 16!
2) 7, 1, and 6 are factors of 16! as they are multiples of 16!, 18 is also a factor of 16! because 2 and 9 are factors of 16!
3) By deduction, the only factor that is not a multiple of 16! is 17, and this means that it is a multiple of 16!+1

Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 6236
Location: United States (CA)
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

01 Feb 2017, 09:23
ankitranjan wrote:
Find the number that divides 16!+1?

A. 7
B. 11
C. 17
D. 6
E. 18

We need to determine which one of the numbers in the given answer choices divides into 16! + 1. In other words, which one is a factor of 16!+1. To determine this, we must recognize that 16! and 16! + 1 are consecutive integers, and consecutive integers will never share the same prime factors. Thus, 16! and 16! + 1 must have different prime factors.

However, rather than breaking 16! factorial into primes, we can look at the answer choices to determine which choice is not a factor of 16!. Since 16! = 16 x 15 x 14…5 x 4 x 3 x 2 x 1, we see that choices A, B, and D are factors of 16! Since 18 = 2 x 9, 2 and 9 are also factors of 16!. However, none of these numbers (6, 7, 11, and 18) will be a factor of 16! + 1, so the only number that can be a factor of 16! + 1 is 17.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Non-Human User
Joined: 09 Sep 2013
Posts: 11012
Re: Find the number that divides 16!+1?  [#permalink]

### Show Tags

08 Mar 2019, 22:56
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: Find the number that divides 16!+1?   [#permalink] 08 Mar 2019, 22:56
Display posts from previous: Sort by