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Which of the following numbers is a perfect square?
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02 Aug 2018, 00:36
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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?
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02 Aug 2018, 02:19
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 numberIf 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 dave13As 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*(n1)! 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?
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02 Aug 2018, 02:34
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 numberIf 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?
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02 Aug 2018, 10:24
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*(n1)! and in your solution +sign Hey dave13I 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?
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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



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Which of the following numbers is a perfect square?
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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 numberIf 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?
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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. \((24!)(25! + 24!)\)
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Re: Which of the following numbers is a perfect square?
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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.
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Re: Which of the following numbers is a perfect square?
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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 numberIf 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



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Re: Which of the following numbers is a perfect square?
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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
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Which of the following numbers is a perfect square?
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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*(n1)! and in your solution +sign



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Re: Which of the following numbers is a perfect square?
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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.
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Re: Which of the following numbers is a perfect square?
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21 Jun 2019, 11:37
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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Re: Which of the following numbers is a perfect square?
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