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If q, r, and s are consecutive even integers and q < r < s, which of

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Yalephd
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If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  06 May 2011, 17:13
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If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s^2 – r^2 – q^2?

A -20
B 0
C 8
D 12
E 16

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-q-r-and-s-are-consecutive-even-integers-and-q-r-s-which-of-t-187235.html
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KarishmaB
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  08 May 2011, 11:30
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Yalephd wrote:
If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s2 – r2 – q2?

A -20
B 0
C 8
D 12
E 16

Is there a quick way to answer a question like this?


Whenever you assume values, try and make them as accurate as possible i.e. I am looking for consecutive even numbers so I will assume the values to be (2n - 2), 2n and (2n + 2)
\(s^2 - r^2 - q^2 = (2n + 2)^2 - (2n)^2 - (2n - 2)^2\)
You get: 4n (4 - n)
So the solution must have 4 as a factor, which we see all options do.
To get -20, n(4-n) must be (-5). It will happen when n = 5.
To get 0, n = 0
To get 12, n(4-n) must be 3. It will happen when n = 3
To get 16, n(4-n) must be 4. It will happen when n = 2
There is no way n(4-n) could be 2 if n is an integer so 8 is not possible.
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patanahi
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  06 May 2011, 18:01
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The three number can be X, X+2, x+4

s^2- r^2 - q^2
(X+4)^2 - (x+2)^2 - x^2
X^2 + 8X + 16 - X^2 - 4X - 4 - X^2
-X^2 - 4x + 12
-(x-6) (x+2)

Answer "C" can not be the value.
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Yalephd
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  06 May 2011, 18:05
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Patanahi, how do you use -(x-6) (x+2) to eliminate answer choices A, B, D & E? Do you have to set -(x-6) (x+2) = to each choice and test them?

patanahi wrote:
The three number can be X, X+2, x+4

s^2- r^2 - q^2
(X+4)^2 - (x+2)^2 - x^2
X^2 + 8X + 16 - X^2 - 4X - 4 - X^2
-X^2 - 4x + 12
-(x-6) (x+2)

Answer "C" can not be the value.
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amit2k9
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  06 May 2011, 22:21
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testing for the blocks -4,-2,0 ; -2,0,2; 0,2,4 and 2,4,6 eliminates -20,0,12 and 16 respectively.

well a quick approach is to check the min and the max values here which are -20 and 16. This will give the range of even numbers to be tested.

I did the same and just kept shifting the window.
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subhashghosh
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  08 May 2011, 08:17
s^2- r^2 - q^2 = (q+4)^2- (q+2)^2 - q^2

= q^2 + 16 + 2*4*q - (q^2 + 2*2q + 4) - q^2

= -q^2 + 12 + 4q

All except 8 fit in this.

Answer - C
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PallavTandon
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  08 May 2011, 11:43
Assuming the numbers to be
2n, 2n+2, 2n+4
then substituting the values if expression s2-r2-q2 gives a quadratic equation which can be simplified as follows

-4(n+1)(n-3)
n=1,2,3,4
16 ,12,0,-20
eliminating 8 as the answer. So choice C
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink] New post  17 Oct 2015, 12:41
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Re: If q, r, and s are consecutive even integers and q < r < s, which of [#permalink]
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