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A perfect square is a number that becomes an integer when square rooti

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New post 10 Apr 2019, 00:51
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A perfect square is a number that becomes an integer when square rooting it. A, B, and C are three positive integers. The ratio of the three numbers is 1 : 2 : 3, respectively. Which one of the following expressions must be a perfect square?


(A) A + B + C

(B) A^2 + B^2 + C^2

(C) A^3 + B^3 + C^3

(D) 3A^2 + B^2 + C^2

(E) 3A^2 + 4B^2 + 4C^2

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A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post Updated on: 10 Apr 2019, 03:32
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Bunuel wrote:
A perfect square is a number that becomes an integer when square rooting it. A, B, and C are three positive integers. The ratio of the three numbers is 1 : 2 : 3, respectively. Which one of the following expressions must be a perfect square?


(A) A + B + C

(B) A^2 + B^2 + C^2

(C) A^3 + B^3 + C^3

(D) 3A^2 + B^2 + C^2

(E) 3A^2 + 4B^2 + 4C^2



let assume the values in the light of ratio.

A = 1
B = 2
C = 3.

option C and D give us integer after square rooting the sum.

plug - 2, 4 , 6.

D is the best answer.

Originally posted by KSBGC on 10 Apr 2019, 00:57.
Last edited by KSBGC on 10 Apr 2019, 03:32, edited 1 time in total.
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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 10 Apr 2019, 01:08
Should be D.

Assume a, b, and c to be 1, 2, & 3 respectively.

Option C and D fit perfectly.

Then assume a, b, and c to be 2, 4, & 6
Only option D makes sense.

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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 10 Apr 2019, 01:16
Let the values of A,B And C be X,2x,3x respectively...

Plug in these values to find out. Only option D turns out to be 16x^2.. which is 4^2*x^2.. which is a perfect square....

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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 10 Apr 2019, 03:17
Bunuel wrote:
A perfect square is a number that becomes an integer when square rooting it. A, B, and C are three positive integers. The ratio of the three numbers is 1 : 2 : 3, respectively. Which one of the following expressions must be a perfect square?


(A) A + B + C

(B) A^2 + B^2 + C^2

(C) A^3 + B^3 + C^3

(D) 3A^2 + B^2 + C^2

(E) 3A^2 + 4B^2 + 4C^2


IMO D ; ratio 1:2:3 so let a,b,c be 1,2,3
test value in D we get = 16 and √ sum of 3A^2 + B^2 + C^2 = 4
IMO D
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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 10 Apr 2019, 06:28
Bunuel is there any other method to solve this? Or we need to solve it just by plugging in numbers? I took over 2mins to solve it that way.

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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 10 Apr 2019, 08:55
sumi747 wrote:
Bunuel is there any other method to solve this? Or we need to solve it just by plugging in numbers? I took over 2mins to solve it that way.

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If you just use x,2x,3x it's really fast and can be done under a minute
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Re: A perfect square is a number that becomes an integer when square rooti  [#permalink]

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New post 13 Apr 2019, 18:32
Bunuel wrote:
A perfect square is a number that becomes an integer when square rooting it. A, B, and C are three positive integers. The ratio of the three numbers is 1 : 2 : 3, respectively. Which one of the following expressions must be a perfect square?


(A) A + B + C

(B) A^2 + B^2 + C^2

(C) A^3 + B^3 + C^3

(D) 3A^2 + B^2 + C^2

(E) 3A^2 + 4B^2 + 4C^2



We can let A = x, B = 2x and C = 3x for some integer x. Let’s analyze each answer choice. We have:

A) A + B + C = x + 2x + 3x = 6x is not necessarily a perfect square.

B) A^2 + B^2 + C^2 = x^2 + 4x^2 + 9x^2 = 13x^2 is not necessarily a perfect square.

C) A^3 + B^3 + C^3 = x^3 + 8x^3 + 27x^3 = 36x^3 is not necessarily a perfect square.

D) 3A^2 + B^2 + C^2 = 3x^2 + 4x^2 + 9x^2 = 16x^2 IS a perfect square since 16x^2 = (4x)^2.

Answer: D
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Re: A perfect square is a number that becomes an integer when square rooti   [#permalink] 13 Apr 2019, 18:32
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