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Math Expert V
Joined: 02 Sep 2009
Posts: 59641
If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 51% (01:48) correct 49% (01:53) wrong based on 51 sessions

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If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z + Y > 0

(2) Z^2 = X^2 + Y^2

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Math Expert V
Joined: 02 Aug 2009
Posts: 8302
Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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1
1
Bunuel wrote:
If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z – Y > 0

(2) Z^2 = X^2 + Y^2

(1) X – Z – Y > 0
$$X – Z – Y > 0......X>Z+Y$$
As All are positive integers, X is greater than SUM of Z and Y, so Surely gtreater than their difference too...X>Z-Y
Suff

(2) Z^2 = X^2 + Y^2
This says that Z is a hypotenuse with X and Y as other two sides.. May not help here
$$X^2=Z^2-Y^2=(Z+Y)(Z-Y)=X*X$$
Surely for this to be true, one of (Z+Y) and (Z-Y) should be GREATER than X and other LESS than Y..
But Sum of Z and Y has to be greater than difference of Z and Y. So, so X<Z+Y, but X>Z-Y.
Suff

D, although statement I says X is the greatest while II says Z is the greatest, but statement II is well written to deduce the answer.
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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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chetan2u wrote:
Bunuel wrote:
If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z – Y > 0

(2) Z^2 = X^2 + Y^2

(1) X – Z – Y > 0
$$X – Z – Y > 0......X>Z+Y$$
As All are positive integers, X is greater than SUM of Z and Y, so Surely gtreater than their difference too...X>Z-Y
Suff

(2) Z^2 = X^2 + Y^2
This says that Z is a hypotenuse with X and Y as other two sides.. May not help here
$$X^2=Z^2-Y^2=(Z+Y)(Z-Y)=X*X$$
Surely for this to be true, one of (Z+Y) and (Z-Y) should be GREATER than X and other LESS than Y..
But Sum of Z and Y has to be greater than difference of Z and Y. So, so X<Z+Y, but X>Z-Y.
Suff

D, although statement I says X is the greatest while II says Z is the greatest, but statement II is well written to deduce the answer.

Edited (1) so that the statements do not contradict. Thank you.
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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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If X, Y and Z are positive integers, is X greater than Z – Y?

Among X, Y and Z which one is greatest or which one is smallest is unknown.

(1) X – Z + Y > 0
Let X = 1, Y = 2 and Z = 3. Then X – Z + Y = 2 > 0
Thus, X = Z - Y. Hence X > Z - Y NO.

Let X = 3, Y = 2 and Z = 1. Then X – Z + Y = 4 > 0
Thus, X > Z - Y. Hence X > Z - Y YES.

INSUFFICIENT.

(2) Z^2 = X^2 + Y^2
This implies that Z is the greatest of all the three positive integers in fact they represent sides of a right angle triangle. So,

X^2 = Z^2 - Y^2
X^2 = (Z - Y) * (Z + Y)
In ether case here X would be such that Z - Y < X < Z + Y. Eg. in the set of 3,4,5 Z = 5 and X takes any value among 3 or 4.

Hence X > Z - Y Always.

SUFFICIENT.

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If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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X, Y and Z are positive integers, is X greater than Z—Y?

Statement1: X—Z+Y> 0
—> X> Z—Y (Always Yes)
Sufficient

Statement2: $$Z^{2}= X^{2} + Y^{2}$$
As chetan2u told, this equality is about right angled triangle (z is a hypotenuse and two other sides of a triangle)
—> according to features of a triangle,
X+Y> Z or
X+Z> Y or
Z+Y> X —>
——————-
X+Y> Z
—> X> Z—Y
(Always Yes)
Sufficient

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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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1
Known: X,Y,Z positive integers

Q. X > Z–Y ? or X-Z+Y > 0 ?

(1) X–Z+Y > 0
This statement directly answers the question.
SUFF

(2) Z^2 = X^2 + Y^2
If X=3, Y=4, Z=5, then X> Z–Y
If X=5, Y=12, Z=13, then X> Z–Y
SUFF

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Senior Manager  P
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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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lnm87 wrote:
If X, Y and Z are positive integers, is X greater than Z – Y?

Among X, Y and Z which one is greatest or which one is smallest is unknown.

(1) X – Z + Y > 0
Let X = 1, Y = 2 and Z = 3. Then X – Z + Y = 2 > 0
Thus, X = Z - Y. Hence X > Z - Y NO.

Let X = 3, Y = 2 and Z = 1. Then X – Z + Y = 4 > 0
Thus, X > Z - Y. Hence X > Z - Y YES.

INSUFFICIENT.

(2) Z^2 = X^2 + Y^2
This implies that Z is the greatest of all the three positive integers in fact they represent sides of a right angle triangle. So,

X^2 = Z^2 - Y^2
X^2 = (Z - Y) * (Z + Y)
In ether case here X would be such that Z - Y < X < Z + Y. Eg. in the set of 3,4,5 Z = 5 and X takes any value among 3 or 4.

Hence X > Z - Y Always.

SUFFICIENT.

Again a silly error. Either all of them are equal X = Y = Z = 1 then X > Z - Y
or such that X – Z + Y > 0. Thus, X > Z - Y always

For foolproof method it is best to say that statement 1 states that one side of a triangle is less than sum of other two sides.
X + Y > Z.

Hence Statement 1 is SUFFICIENT.
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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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chetan2u wrote:
Bunuel wrote:
If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z – Y > 0

(2) Z^2 = X^2 + Y^2

(1) X – Z – Y > 0
$$X – Z – Y > 0......X>Z+Y$$
As All are positive integers, X is greater than SUM of Z and Y, so Surely gtreater than their difference too...X>Z-Y
Suff

(2) Z^2 = X^2 + Y^2
This says that Z is a hypotenuse with X and Y as other two sides.. May not help here
$$X^2=Z^2-Y^2=(Z+Y)(Z-Y)=X*X$$
Surely for this to be true, one of (Z+Y) and (Z-Y) should be GREATER than X and other LESS than Y..
But Sum of Z and Y has to be greater than difference of Z and Y. So, so X<Z+Y, but X>Z-Y.
Suff

D, although statement I says X is the greatest while II says Z is the greatest, but statement II is well written to deduce the answer.

Beautiful solution for statement II
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Re: If X, Y and Z are positive integers, is X greater than Z – Y?  [#permalink]

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Bunuel wrote:
If X, Y and Z are positive integers, is X greater than Z – Y?

(1) X – Z + Y > 0

(2) Z^2 = X^2 + Y^2

(x,y,z) = positive integers

$$x>z-y…x-z+y>0$$

(1) X – Z + Y > 0 sufic

(2) Z^2 = X^2 + Y^2 sufic

$$z^2=x^2+y^2…x^2=z^2-y^2…(x=positive.int)…x^2>0…z^2-y^2>0…z>y$$
$$x>z-y…x^2>(z-y)^2…(z^2-y^2)>(z^2+y^2-2zy)…-2y^2>-2zy…y<z$$

Ans (D) Re: If X, Y and Z are positive integers, is X greater than Z – Y?   [#permalink] 19 Nov 2019, 06:44
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