NEW HERE? LEARN MORE
closeclose
Last visit was: 24 Apr 2024, 22:02 It is currently 24 Apr 2024, 22:02
Decision Tracker
My Rewards
New posts
New comers' posts

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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

Which of the following numbers is a perfect square?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92900
Own Kudos [?]: 618845 [6]
Given Kudos: 81588
Send PM
CAT Tests Forum Quiz Mode
Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 00:36
6
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

45% (medium)

Question Stats:

63% (01:20) correct 37% (01:36) wrong based on 168 sessions
Hide Show timer Statistics
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)
ShowHide Answer
Official Answer
D
User avatar
L
pushpitkc
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 2873
Own Kudos [?]: 5205 [1]
Given Kudos: 47
Location: India
GPA: 3.12
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 02:19
1
Kudos
dave13 wrote:
pushpitkc wrote:
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)



A factorial can be written as the product of that number and the factorial of the smaller number
If you look at the numbers in the answer options, \(25(5^2)\) is a number which is a
perfect square! We need to backtrack from answer options and arrive at a 25.

\((23!)(24! + 23!) = (23!)(24*23! + 23!) = (23!)^2(24 + 1) = (23!)^2*25 = (23! * 5)^2\)

Therefore, \((23!)(24! + 23!)\)(Option D) is a perfect square and is our answer!


hi there pushpitkc :-)
how did you get that the result is a perfect square \((23!)(24! + 23!)\)
should try all answer choices as you did ? isnt it time consuming ? or perhaps there is a way you can quickly filter incorrect answer choices ?
thanks for taking time to explain :-)


Hi dave13

As I have already explained - After seeing the various options, I knew that
25 is a number which is both a perfect square and can be got using the five
answer options. Also, see the highlighted part in my solution

The general rule for the highlighted part is n! = n*(n-1)! Specific to the
problem in hand, 24! = 24*23! and 23! + 24! = 23!(1 + 24) = 23!*25

Hope this clears your confusion.
User avatar
G
CounterSniper
Director
Director
Joined: 20 Feb 2015
Posts: 631
Own Kudos [?]: 712 [1]
Given Kudos: 74
Concentration: Strategy, General Management
Send PM
Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 02:34
1
Kudos
dave13 wrote:
pushpitkc wrote:
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)



A factorial can be written as the product of that number and the factorial of the smaller number

If you look at the numbers in the answer options, \(25(5^2)\) is a number which is a
perfect square! We need to backtrack from answer options and arrive at a 25.

\((23!)(24! + 23!) = (23!)(24*23! + 23!) = (23!)^2(24 + 1) = (23!)^2*25 = (23! * 5)^2\)

Therefore, \((23!)(24! + 23!)\)(Option D) is a perfect square and is our answer!


hi there pushpitkc :-)
how did you get that the result is a perfect square \((23!)(24! + 23!)\)
should try all answer choices as you did ? isnt it time consuming ? or perhaps there is a way you can quickly filter incorrect answer choices ?
thanks for taking time to explain :-)



not sure if this will help !!

23! will definitely be an integer say p
now, p*p = p^2
this part is a perfect square , as it gives p as the root
the remaining is 25 , which again is 5*5 and gives 5 as its root

also, perfect square * perfect square = perfect square
User avatar
L
pushpitkc
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 2873
Own Kudos [?]: 5205 [1]
Given Kudos: 47
Location: India
GPA: 3.12
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 10:24
1
Kudos
dave13 wrote:
hi pushpitkc thanks got it. just one tech question :-)

how from here \((23!)(24*23! + 23!) How You Got This (23!)^2(24 + 1)\) ?

here \((23!)(24*23! + 23!)\) I see TWO 23! in brackets plus ONE 23! outside of brackets so normally it must be \((23!)^3(24 + 1)\) when you factor out :? pls explain :)

Also why here in formula there is minus sign n! = n*(n-1)! and in your solution +sign :?


Hey dave13

I think Bunuel explained this perfectly - while explaining a similar expression's specification

Let's assume we have an expression -> c(ab + a) | This can be further simplified as ca(b+1).
In the example that we are given (23!)(24*23! + 23!) = (23!)^2 * (24 + 1)

Hope this helps you!
User avatar
G
CounterSniper
Director
Director
Joined: 20 Feb 2015
Posts: 631
Own Kudos [?]: 712 [0]
Given Kudos: 74
Concentration: Strategy, General Management
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 00:45
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)



should be D

23!*23!(24+1)
23!*23!*25
User avatar
L
pushpitkc
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 2873
Own Kudos [?]: 5205 [0]
Given Kudos: 47
Location: India
GPA: 3.12
Send PM
Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 00:47
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)



A factorial can be written as the product of that number and the factorial of the smaller number

If you look at the numbers in the answer options, \(25(5^2)\) is a number which is a
perfect square! We need to backtrack from answer options and arrive at a 25.

\((23!)(24! + 23!) = (23!)(24*23! + 23!) = (23!)^2(24 + 1) = (23!)^2*25 = (23! * 5)^2\)

Therefore, \((23!)(24! + 23!)\)(Option D) is a perfect square and is our answer!
User avatar
G
D3N0
Senior Manager
Senior Manager
Joined: 21 Jan 2015
Posts: 423
Own Kudos [?]: 356 [0]
Given Kudos: 82
Location: India
Concentration: Strategy, Marketing
Schools: Smeal'21 (WD) Arizona State'21 (A$) Simon '21 (WD)
GMAT 1: 620 Q48 V28
GMAT 2: 690 Q49 V35
WE:Sales (Consumer Products)
Send PM
GMAT ToolKit User Reviews Badge
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 00:48
Ans: D
given expression looks a bit ... you know what i mean: so lets make it easy lets say that first number in ! is x so expression will become
\((x!)((x+1)! + x!)\) from here.. let me also write !x as F(x)
If we take the F(x) inside, expression will become [F(x)*F(x+1) + F(x)^2]
We know F(x+1) = (x+1)*F(x)
so it becomes [F(x)^2 * (x+1) + F(x)^2]
npw take F(x)^2 common Expression will become = F(x)^2 [x+1+1]
F(x)^2 [x+2]
now we need to know if this is whole square or not. We know F(x)^2 is always so we just need to know if (x+2) is whole square or not.
Put x =23 , we know 25 is a whole square so D is the ans.

Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)
User avatar
P
sudarshan22
Retired Moderator
Joined: 30 Jan 2015
Posts: 636
Own Kudos [?]: 2427 [0]
Given Kudos: 1131
Location: India
Concentration: Operations, Marketing
GPA: 3.5
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 02:05
Bunuel wrote:
Which of the following numbers is a perfect square?

+1 for D.

(23!)(24!+23!)
23! * ( 24 * 23! + 23! )
23! * 23! * ( 24 + 1 )
23! * 23! * 25
(23!)^2 * 5^2

Hence, D.
User avatar
L
dave13
VP
VP
Joined: 09 Mar 2016
Posts: 1160
Own Kudos [?]: 1017 [0]
Given Kudos: 3851
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 02:10
pushpitkc wrote:
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)



A factorial can be written as the product of that number and the factorial of the smaller number

If you look at the numbers in the answer options, \(25(5^2)\) is a number which is a
perfect square! We need to backtrack from answer options and arrive at a 25.

\((23!)(24! + 23!) = (23!)(24*23! + 23!) = (23!)^2(24 + 1) = (23!)^2*25 = (23! * 5)^2\)

Therefore, \((23!)(24! + 23!)\)(Option D) is a perfect square and is our answer!


hi there pushpitkc :-)
how did you get that the result is a perfect square \((23!)(24! + 23!)\)
should try all answer choices as you did ? isnt it time consuming ? or perhaps there is a way you can quickly filter incorrect answer choices ?
thanks for taking time to explain :-)
User avatar
B
EvanSamPhuong
Intern
Intern
Joined: 24 Jul 2018
Posts: 4
Own Kudos [?]: 0 [0]
Given Kudos: 15
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 03:01
Ans: D

I saw a pattern in the answer choices and formularized it:

n![(n+1)!+n!]

Then, I factored it further:

n!*n!*(n+1+1)
= [(n!)^2]*(n+2)

=> n+2 has to be a perfect square

Judging from answer choices, n=23

Posted from my mobile device
User avatar
L
dave13
VP
VP
Joined: 09 Mar 2016
Posts: 1160
Own Kudos [?]: 1017 [0]
Given Kudos: 3851
Send PM
Which of the following numbers is a perfect square? [#permalink] New post   02 Aug 2018, 07:17
hi pushpitkc thanks got it. just one tech question :-)

how from here \((23!)(24*23! + 23!) How You Got This (23!)^2(24 + 1)\) ?

here \((23!)(24*23! + 23!)\) I see TWO 23! in brackets plus ONE 23! outside of brackets so normally it must be \((23!)^3(24 + 1)\) when you factor out :? pls explain :)

Also why here in formula there is minus sign n! = n*(n-1)! and in your solution +sign :?
User avatar
P
LeoN88
BSchool Moderator
Joined: 08 Dec 2013
Posts: 686
Own Kudos [?]: 515 [0]
Given Kudos: 227
Location: India
Concentration: Nonprofit, Sustainability
Schools: ISB '23
GMAT 1: 630 Q47 V30
WE:Operations (Non-Profit and Government)
Send PM
Reviews Badge
Re: Which of the following numbers is a perfect square? [#permalink] New post   18 Jun 2019, 18:06
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)


When I see the term perfect square then the first thing that comes to my mind is that all the numbers should occur twice in a series of a multiplication chain.
I see addition, I have to remove this sign as sum of two perfect squares is not a perfect square (necessarily), (9+4)
I also see that the numbers are revolving around mid twenties, so GMAT is toying with 25 that is a perfect square.

If I look at option D, I can easily eliminate the '+' sign
23! * 23! (24+1)

Perfect square spotted.
User avatar
L
ScottTargetTestPrep
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 18756
Own Kudos [?]: 22049 [0]
Given Kudos: 283
Location: United States (CA)
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   21 Jun 2019, 11:37
Expert Reply
Bunuel wrote:
Which of the following numbers is a perfect square?


A. \((20!)(21! + 20!)\)

B. \((21!)(22! + 21!)\)

C. \((22!)(23! + 22!)\)

D. \((23!)(24! + 23!)\)

E. \((24!)(25! + 24!)\)


In order for a number to be a perfect square, we need our factors to be in even quantities.

Looking at answer choice D, we see that we have:

23![23!(24 + 1)]

23![23!(25)]

(23!)^2 x 5^2

Thus, the quantity in choice D is a perfect square.

Answer: D
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32657
Own Kudos [?]: 821 [0]
Given Kudos: 0
Send PM
Re: Which of the following numbers is a perfect square? [#permalink] New post   03 Feb 2024, 22:45
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.
GMAT Club Bot
gmatclubot
Re: Which of the following numbers is a perfect square? [#permalink]
03 Feb 2024, 22:45
Moderators:
Bunuel
Math Expert
92900 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions