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Math Expert V
Joined: 02 Sep 2009
Posts: 57074
Which of the following numbers is a perfect square?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 63% (01:23) correct 37% (01:38) wrong based on 113 sessions

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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. $$(24!)(25! + 24!)$$

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Re: Which of the following numbers is a perfect square?  [#permalink]

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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.
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Which of the following numbers is a perfect square?  [#permalink]

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1
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
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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!
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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
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Which of the following numbers is a perfect square?  [#permalink]

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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!
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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. $$(24!)(25! + 24!)$$

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Re: Which of the following numbers is a perfect square?  [#permalink]

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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.
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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 Intern  B
Joined: 24 Jul 2018
Posts: 8
Re: Which of the following numbers is a perfect square?  [#permalink]

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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

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Which of the following numbers is a perfect square?  [#permalink]

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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 ISB School Moderator G
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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.
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Re: Which of the following numbers is a perfect square?  [#permalink]

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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
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