Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 20 Jul 2019, 16:00

### 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

# x# is defined for every positive even integer x as the product of all

Author Message
TAGS:

### Hide Tags

Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

06 Sep 2015, 10:28
3
7
00:00

Difficulty:

55% (hard)

Question Stats:

55% (01:37) correct 45% (01:25) wrong based on 355 sessions

### HideShow timer Statistics

x# is defined for every positive even integer x as the product of all even integers from 2 to x. What is the smallest possible prime factor of (x#+7)?

A. 2
B. 3
C. 5
D. 7
E. 11

_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
Intern
Joined: 16 Aug 2015
Posts: 18
x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

Updated on: 06 Sep 2015, 17:41
1
We can resolve this problem by picking some smart numbers.

First of all, x# is an even number, accordingly x#+7 is odd, so the answer cannot be 2 even if it's tempting at the first sight !
Now we can proceed by taking some numbers, if we consider

X=4, then x#+7= 15 = 3 x 5. we got it, 3 is the smallest prime factor that we can obtain.

Originally posted by issamL on 06 Sep 2015, 14:28.
Last edited by issamL on 06 Sep 2015, 17:41, edited 1 time in total.
Manager
Joined: 14 Mar 2014
Posts: 145
GMAT 1: 710 Q50 V34
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

06 Sep 2015, 17:23
1
IMO: B

Since smallest possible prime factor is required.
Let x = 2 then --> 2# = 2
x#+7 ==> 2#+7 = 2+7 = 9 = 3*3.
Thus 3 is the smallest possible Prime Factor

2 cannot be the possible prime factor because --> x# is always even since it is product of even numbers
Even +7 = Odd. Hence 2 cannot be prime factor of this result.
_________________
I'm happy, if I make math for you slightly clearer
And yes, I like kudos
¯\_(ツ)_/¯
Current Student
Joined: 08 Jan 2015
Posts: 80
Location: Thailand
GMAT 1: 540 Q41 V23
GMAT 2: 570 Q44 V24
GMAT 3: 550 Q44 V21
GMAT 4: 660 Q48 V33
GPA: 3.31
WE: Science (Other)
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

14 Sep 2015, 21:32
issamL wrote:
We can resolve this problem by picking some smart numbers.

First of all, x# is an even number, accordingly x#+7 is odd, so the answer cannot be 2 even if it's tempting at the first sight !
Now we can proceed by taking some numbers, if we consider

X=4, then x#+7= 15 = 3 x 5. we got it, 3 is the smallest prime factor that we can obtain.

If x=4, should x#+7 be 31?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9446
Location: Pune, India
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

14 Sep 2015, 23:34
1
Aves wrote:
issamL wrote:
We can resolve this problem by picking some smart numbers.

First of all, x# is an even number, accordingly x#+7 is odd, so the answer cannot be 2 even if it's tempting at the first sight !
Now we can proceed by taking some numbers, if we consider

X=4, then x#+7= 15 = 3 x 5. we got it, 3 is the smallest prime factor that we can obtain.

If x=4, should x#+7 be 31?

No. # is a user defined operator i.e. the question defines for what # stand.

x# = 2*4*..*x

So if x = 2, x# = 2
If x = 4, x# = 2*4 = 8
If x = 6, x# = 2*4*6 = 48
and so on...

So x# + 7 when x = 4 will be 8 + 7 = 15 because when x is 4, x# is 8.
_________________
Karishma
Veritas Prep GMAT Instructor

Current Student
Joined: 08 Jan 2015
Posts: 80
Location: Thailand
GMAT 1: 540 Q41 V23
GMAT 2: 570 Q44 V24
GMAT 3: 550 Q44 V21
GMAT 4: 660 Q48 V33
GPA: 3.31
WE: Science (Other)
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

14 Sep 2015, 23:45
VeritasPrepKarishma wrote:
Aves wrote:
issamL wrote:
We can resolve this problem by picking some smart numbers.

First of all, x# is an even number, accordingly x#+7 is odd, so the answer cannot be 2 even if it's tempting at the first sight !
Now we can proceed by taking some numbers, if we consider

X=4, then x#+7= 15 = 3 x 5. we got it, 3 is the smallest prime factor that we can obtain.

If x=4, should x#+7 be 31?

No. # is a user defined operator i.e. the question defines for what # stand.

x# = 2*4*..*x

So if x = 2, x# = 2
If x = 4, x# = 2*4 = 8
If x = 6, x# = 2*4*6 = 48
and so on...

So x# + 7 when x = 4 will be 8 + 7 = 15 because when x is 4, x# is 8.

Oh, it was only the product of all even integers not all integers.

Intern
Joined: 01 Sep 2016
Posts: 6
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

18 Dec 2016, 20:50
When x is taken as 6,X#+7 will be 55.
In this case ,the smallest prime factor will be 5 right?
Senior Manager
Joined: 13 Oct 2016
Posts: 363
GPA: 3.98
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

19 Dec 2016, 02:57
1
reto wrote:
x# is defined for every positive even integer x as the product of all even integers from 2 to x. What is the smallest possible prime factor of (x#+7)?

A. 2
B. 3
C. 5
D. 7
E. 11

We have

$$2*4*6*8*10* … * x = 2^{\frac{x}{2}}*1*2*3*4*5* … * \frac{x}{2} = 2^{\frac{x}{2}}*(\frac{x}{2})!$$

If we need this expression to be an integer, min value that $$x$$ can take is $$2$$.

$$x# + 7 = 2^1*1! + 7 = 9 = 3^2$$

Min prime factor is $$3$$

Director
Joined: 02 Sep 2016
Posts: 657
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

03 Apr 2017, 23:58
x# is the product of all even nos. from 2 to x. Thus the answer would be an even no.

x#+7= Even +Odd= Odd

Therefore the smallest possible prime factor would be 3 and not 2 because the final answer is ODD and not even.
_________________
Help me make my explanation better by providing a logical feedback.

If you liked the post, HIT KUDOS !!

Don't quit.............Do it.
Intern
Joined: 07 Dec 2016
Posts: 38
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

10 Apr 2017, 05:43
Yes it will have to be odd and thus 3 is the small prime factor
_________________
Cheers!
If u like my post..... payback in Kudos!!
Director
Joined: 02 Sep 2016
Posts: 657
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

13 Apr 2017, 09:35
VeritasPrepKarishma wrote:
Aves wrote:
issamL wrote:
We can resolve this problem by picking some smart numbers.

First of all, x# is an even number, accordingly x#+7 is odd, so the answer cannot be 2 even if it's tempting at the first sight !
Now we can proceed by taking some numbers, if we consider

X=4, then x#+7= 15 = 3 x 5. we got it, 3 is the smallest prime factor that we can obtain.

If x=4, should x#+7 be 31?

No. # is a user defined operator i.e. the question defines for what # stand.

x# = 2*4*..*x

So if x = 2, x# = 2
If x = 4, x# = 2*4 = 8
If x = 6, x# = 2*4*6 = 48
and so on...

So x# + 7 when x = 4 will be 8 + 7 = 15 because when x is 4, x# is 8.

But karishma (48+7= 55) which is not divisible by 3.
So it is not true in all cases.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9446
Location: Pune, India
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

13 Apr 2017, 22:59
Shiv2016 wrote:

But karishma (48+7= 55) which is not divisible by 3.
So it is not true in all cases.

We need the smallest "possible" prime factor which CAN be a factor of a number of the form x# + 7.

There are some prime factors which CANNOT be factors of a number of this form. One such prime factor is 2 because numbers of the form x# + 7 will always be odd.

The next smallest prime factor is 3. Since there is a number of the form x# + 7 for which 3 is a factor (it needn't be a factor of EVERY such number), 3 is the smallest such prime factor.
_________________
Karishma
Veritas Prep GMAT Instructor

Director
Joined: 02 Sep 2016
Posts: 657
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

13 Apr 2017, 23:36
VeritasPrepKarishma wrote:
Shiv2016 wrote:

But karishma (48+7= 55) which is not divisible by 3.
So it is not true in all cases.

We need the smallest "possible" prime factor which CAN be a factor of a number of the form x# + 7.

There are some prime factors which CANNOT be factors of a number of this form. One such prime factor is 2 because numbers of the form x# + 7 will always be odd.

The next smallest prime factor is 3. Since there is a number of the form x# + 7 for which 3 is a factor (it needn't be a factor of EVERY such number), 3 is the smallest such prime factor.

Okay. Got it. Thanks!!
Intern
Joined: 13 Feb 2018
Posts: 13
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

15 Jul 2018, 04:46
Any conceptual approach to solving this problem?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9446
Location: Pune, India
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

16 Jul 2018, 04:00
1
agarwalaayush2007 wrote:
Any conceptual approach to solving this problem?

x# is defined for every positive even integer x as the product of all even integers from 2 to x.
implies
2# = 2
4# = 2*4 = 8
6# = 2*4*6 = 48
and so on...

What is the smallest possible prime factor of (x#+7)?
2# + 7 will be 9
4# + 7 will be 15
6# + 7 will be 55
and so on...

Note that x# will always be even so if you add 7 to it, it will become odd. So 2 will never be a factor of x# + 7.
Next we see that 3 CAN be a factor of x# + 7 (9 and 15 have 3 as a factor) so the smallest such prime number must be 3.
_________________
Karishma
Veritas Prep GMAT Instructor

Intern
Joined: 13 Feb 2018
Posts: 13
Re: x# is defined for every positive even integer x as the product of all  [#permalink]

### Show Tags

17 Jul 2018, 07:08
KarishmaB wrote:
agarwalaayush2007 wrote:
Any conceptual approach to solving this problem?

x# is defined for every positive even integer x as the product of all even integers from 2 to x.
implies
2# = 2
4# = 2*4 = 8
6# = 2*4*6 = 48
and so on...

What is the smallest possible prime factor of (x#+7)?
2# + 7 will be 9
4# + 7 will be 15
6# + 7 will be 55
and so on...

Note that x# will always be even so if you add 7 to it, it will become odd. So 2 will never be a factor of x# + 7.
Next we see that 3 CAN be a factor of x# + 7 (9 and 15 have 3 as a factor) so the smallest such prime number must be 3.

Thank you Karishma. Great help!
Re: x# is defined for every positive even integer x as the product of all   [#permalink] 17 Jul 2018, 07:08
Display posts from previous: Sort by