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The operation [symbol] is defined for all integers x a

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The operation [symbol] is defined for all integers x a [#permalink] New post 24 Jan 2016, 15:08
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The operation \(\otimes\) is defined for all integers x and y as \(x \otimes y = xy - y\). If x and y are positive integers, which of the following CANNOT be zero?

A) \(x \otimes y\)

B) \(y \otimes x\)

C) \((x-1)\otimes y\)

D) \((x+1) \otimes y\)

E) \(x \otimes (y-1)\)

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Question: 22
Page: 463
Difficulty: medium
[Reveal] Spoiler: OA

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Last edited by Carcass on 29 Jan 2020, 18:13, edited 2 times in total.
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Re: The operation * is defined for all integers x and y as x * [#permalink] New post 24 Jan 2016, 15:17
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Solution

Note Our symbol can be \(*\) or \($\). It does not matter, it is just a placeholder

The question stem says to us that X and Y are positive integers and that the symbol is defined for all integers . So, the best way to tackle the question is picking numbers

X=1 and Y=2

Scanning and substituting in all the answer choices you can reach the correct answer. For D : our (x+1) is = in to our equation to \(XY\) so we do have that \(1*2=2\). Then, \(X*Y=XY-Y\) following that (\(2+1)-2=1\).

The correct answer is \(D\)

PS: substitute the values in the other answer choices and you will get zero or not. We are searching an answer that CANNOT be zero
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Re: The operation * is defined for all integers x and y as x * [#permalink] New post 14 Jul 2016, 14:15
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Carcass wrote:
The operation * is defined for all integers x and y as x * y = xy − y. If x and  y are positive integers, which of the following CANNOT be zero?

A) X*Y

B) Y*X

C) (X-1)*Y

D) (X+1)*Y

E) X*(Y-1)



Let's take the formula x * y = xy − y, and rewrite is as x * y = y(x − 1)
Now let's check each answer choice (BEGINNING WITH E, since the test-makers like to place the correct answer
for these questions near the end, since most test-takers will check the answers from A to E.)

E) X*(Y-1)
Apply the formula to get: (Y-1)(X-1)
Can this expression ever equal 0?
Sure, if Y = 1 and X = 1, then (Y-1)(X-1) = (1-1)(1-1) = 0
ELIMINATE E

D) (X+1)*Y
Apply the formula to get: (Y)(X+1-1)
Simplify to get (Y)(X)
Can this expression ever equal 0?
NO.
If X and  Y are positive integers, then (Y)(X) can NEVER equal zero

Answer:
[Reveal] Spoiler:
D

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Re: The operation * is defined for all integers x and y as x * [#permalink] New post 26 Oct 2018, 10:52
why E is also not a choice.. I am getting that positive as well.please help
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Re: The operation * is defined for all integers x and y as x * [#permalink] New post 27 Oct 2018, 02:49
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The operation * is defined for all integers x and y as \(x * y = xy - y\). If x and y are positive integers, which of the following CANNOT be zero?

A) \(X*Y\)......XY-Y=0.....Y(X-1)=0.....\(Y\neq{0}\) but X can be 1... possible

B) \(Y*X\)......XY-X=0.....X(Y-1)=0.....\(X\neq{0}\) but Y can be 1... possible

C) \((X-1)*Y\)......(X-1)Y-Y=Y(X-1-1)=Y(X-2)=0.....\(Y\neq{0}\) but X can be 2... possible

D) \((X+1)*Y\)......(X+1)Y-Y=Y(X+1-1)=XY=0.....both X and Y are positive,so \(XY\neq{0}\).... Not possible

E) \(X*(Y-1)\)......X(Y-1)-(Y-1)=(Y-1)(X-1)=0.....any one or both of Y and X can be 1... possible

D
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: The operation * is defined for all integers x and y as x * [#permalink] New post 27 Oct 2018, 03:00
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Reetika1990 wrote:
why E is also not a choice.. I am getting that positive as well.please help

So x*(y-1)=x(y-1)-(y-1)..
Now let x =2 and y =1..2(1-1)-(1-1)=0-0=0
Or x can be anything positove and y =1 ans is 0
Also when y is anything positive and x=1..
x(y-1)-(y-1)=1*(y-1)-(y-1)=(y-1)-(y-1)=0..

So E is possible
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: The operation * is defined for all integers x and y as x *   [#permalink] 27 Oct 2018, 03:00
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