Problem Solving

ziyuenlau wrote:For consecutive integers x, y, and z, where x > y > z, which of the following CANNOT be the value of (x² - y²)(y² - z²)?

(A) 63

(B) 99

(C) 195

(D) 276

(E) 323

The keyword here is CONSECUTIVE INTEGERS.
Notice that consecutive integers alternate ODD, EVEN, ODD, EVEN, ODD, EVEN, ....

Also notice that we can factor the given expression as follows:
(x² - y²)(y² - z²) = (x + y)(x - y)(y + z)(y - z)

Since x and y are consecutive integers, one must be ODD and one must be EVEN
This means that (x + y) is ODD and (x - y) is ODD

Likewise, y and z are consecutive integers, one must be ODD and one must be EVEN
This means that (y + z)is ODD and (y - z) is ODD

So, (x² - y²)(y² - z²) = (x + y)(x - y)(y + z)(y - z)
= (ODD)(ODD)(ODD)(ODD)
= ODD

So, the expression must evaluate to be odd.

Answer: [spoiler]D (since 276 is EVEN)[/spoiler]

Cheers,
Brent

Hi ziyuenlau,

GMAT questions are almost always built around patterns - even if you don't realize that the pattern is there, you can probably do a bit of 'brute force' work and define the pattern. By extension, if you know the pattern, then you should be able to use that knowledge to your advantage to either answer the question immediately (or do another step or two of work to get the answer).

Here, we're given some specific facts to work with:
1) X, Y and Z are CONSECUTIVE integers
2) X > Y > Z

We're asked for what CANNOT be the value of (X^2 - Y^2)(Y^2 - Z^2).

Let's TEST VALUES and see if a pattern emerges...

IF... X = 3, Y = 2, Z = 1....
(9 - 4)(4 - 1) = (5)(3) = 15

So "15" is a possible answer. Also note that we ended up multiplying two ODD numbers together... Let's try another TEST....

IF... X = 4, Y = 3, Z = 2....
(16 - 9)(9 - 4) = (7)(5) = 35

So "35" is a possible answer. Notice that we again ended up multiplying two ODD numbers together... That looks like a pattern. If the end result is just going to be an ODD number every time, then there's clearly an answer that CANNOT be the value...

If you're not convinced yet, then try another example (and feel free to try as many as you like - as the numbers increase, you'll eventually hit all 4 of the possible answers, at which point you'll know which answer is NOT possible.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

You can always represent consecutive integers algebraically, because they have a known relationship:
If x, y, and z are consecutive integers where x > y > z, then:
y = z + 1
x = z + 2

So you could rephrase: (x² - y²)(y² - z²)
((z + 2)² - (z + 1)²)((z + 1)² - z²)
((z² + 4z + 4) - (z² + 2z + 1))((z² + 2z + 1) - z²)
(2z + 3)(2z + 1)
4z² + 8z + 3

4z² and 8z will always be even, so 4z² + 8z + 3 will always be odd.

That said, it would be easier to just think in terms of EVEN and ODD from the beginning, as other posters have pointed out.

Hi hoppycat,

Before we can discuss how you might speed up on these types of questions, it would help to know what you actually did for those 3 minutes. What was your first TEST case? What other examples did you try? How much of your work did you write down (and how much did you do 'in your head?')? Did you recognize that the end calculation was ALWAYS an ODD number?

GMAT assassins aren't born, they're made,
Rich

hazelnut01 wrote:For consecutive integers x, y, and z, where x > y > z, which of the following CANNOT be the value of ( x^2 - y^2 ) ( y^2 - z^2)?

(A) 63

(B) 99

(C) 195

(D) 276

(E) 323

Source : Veritas Prep
OA=D

Very good solutions by experts. One more from my side.

We know that x, y and are consecutive integers such that x > y > z.

By looking at the expression (x^2 - y^2)*(y^2 - z^2), we see that the expression can have many possible values. Secondly, if you scan the options, you find that only one option is even, which is option D. So let's think in that direction.

If z is odd, then y is even and x is odd.

=> (x^2 - y^2)*(y^2 - z^2) = (Odd^2 - Even^2)*(Even^2 - Odd^2)

=> (Odd - Even)*(Even - Odd) = Odd*Odd = Odd

If z is even, then y is odd and x is even.

=> (x^2 - y^2)*(y^2 - z^2) = (Even^2 - Odd^2)*(Odd^2 - Even^2)

=> (Even - Odd)*(Odd - Even) = Odd*Odd = Odd

Thus, in each case, the resultant value is ODD, or option D, 276 is not a possible value.

The correct answer: D

Hope this helps!

-Jay
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hazelnut01 wrote:For consecutive integers x, y, and z, where x > y > z, which of the following CANNOT be the value of ( x^2 - y^2 ) ( y^2 - z^2)?

(A) 63

(B) 99

(C) 195

(D) 276

(E) 323

Source : Veritas Prep
OA=D

Very good solutions by experts. One more from my side.

We know that x, y and are consecutive integers such that x > y > z.

By looking at the expression (x^2 - y^2)*(y^2 - z^2), we see that the expression can have many possible values. Secondly, if you scan the options, you find that only one option is even, which is option D. So let's think in that direction.

If z is odd, then y is even and x is odd.

=> (x^2 - y^2)*(y^2 - z^2) = (Odd^2 - Even^2)*(Even^2 - Odd^2)

=> (Odd - Even)*(Even - Odd) = Odd*Odd = Odd

If z is even, then y is odd and x is even.

=> (x^2 - y^2)*(y^2 - z^2) = (Even^2 - Odd^2)*(Odd^2 - Even^2)

=> (Even - Odd)*(Odd - Even) = Odd*Odd = Odd

Thus, in each case, the resultant value is ODD, or option D, 276 is not a possible value.

The correct answer: D

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Jakarta | Nanjing | Berlin | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.