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# Which of the following numbers is a perfect square?

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

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23 Apr 2019, 12:58
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Difficulty:

15% (low)

Question Stats:

75% (01:05) correct 25% (01:07) wrong based on 52 sessions

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Which of the following numbers is a perfect square?

(A) $$\frac{14!15!}{2}$$

(B) $$\frac{15!16!}{2}$$

(C) $$\frac{16!17!}{2}$$

(D) $$\frac{17!18!}{2}$$

(E) $$\frac{18!19!}{2}$$

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

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Updated on: 23 Apr 2019, 13:13
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Bunuel wrote:
Which of the following numbers is a perfect square?

(A) $$\frac{14!15!}{2}$$

(B) $$\frac{15!16!}{2}$$

(C) $$\frac{16!17!}{2}$$

(D) $$\frac{17!18!}{2}$$

(E) $$\frac{18!19!}{2}$$

Ans should be D.
Option D can be rewritten as \frac{17!* 17!* 18}{2} = $$(17!)^2 * 9$$ = $$(17)!^2*(3)^2$$ which is a perfect square
The same can't be done for rest of the options

Originally posted by manu11 on 23 Apr 2019, 13:08.
Last edited by manu11 on 23 Apr 2019, 13:13, edited 1 time in total.
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Re: Which of the following numbers is a perfect square?  [#permalink]

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23 Apr 2019, 13:12
IMO D

express the options in the form :
17!*18!/2= 17!*17!*18/2=(17!)²*9=(17!)²*(3)²

it not possible for other option provided.

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

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18 Jun 2019, 18:09
Bunuel wrote:
Which of the following numbers is a perfect square?

(A) $$\frac{14!15!}{2}$$

(B) $$\frac{15!16!}{2}$$

(C) $$\frac{16!17!}{2}$$

(D) $$\frac{17!18!}{2}$$

(E) $$\frac{18!19!}{2}$$

The idea is to somehow eliminate two in the denominator (Primary)
Secondary objective is to make sure we have all the numbers multiplied twice to get a perfect square.

Clearly D
I can eliminate 2 and bring a perfect square 9, also 17!*17! will give me perfect square.
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Re: Which of the following numbers is a perfect square?  [#permalink]

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21 Jun 2019, 16:31
Bunuel wrote:
Which of the following numbers is a perfect square?

(A) $$\frac{14!15!}{2}$$

(B) $$\frac{15!16!}{2}$$

(C) $$\frac{16!17!}{2}$$

(D) $$\frac{17!18!}{2}$$

(E) $$\frac{18!19!}{2}$$

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

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

(17!18!)/2 = (17!17!18)/2 = (17!)^2 x 18/2 = (17!)^2 x 3^2

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

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Re: Which of the following numbers is a perfect square?   [#permalink] 21 Jun 2019, 16:31
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