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Intern  Joined: 12 Dec 2013
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If x and y are positive integers, is x a prime number?  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 35% (02:41) correct 65% (02:43) wrong based on 860 sessions

### HideShow timer Statistics If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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7
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

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GMAT 1: 700 Q51 V32 Re: If x and y are positive integers, is x a prime number?  [#permalink]

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6
1
Another way to solve it:

Positive integers are: 1, 2,3, etc

1) |x−2|<2−y
RHS should be greater than 0 as LHS is MOD

Therefore, 2-y > 0
or y<2
or $$y = 1$$ (As y is a +ve integer)
Hence,
|x−2| <2−y
or |x−2|<1
or |x−2|= 0 (As LHS cannot be negative)
x = 2

Hence, Statement 1) is alone sufficient

2) x+y−3=|1−y|

Can be written as:
x+y-3 =|y−1| (Same as it is modulus)
or x - 2 + (y -1) = |y−1|

Using,
X + Y = |Y| Where x-2 = X (Can have Values: -1, 0, 1, 2, etc)
And y -1 = Y (Can have Values: 0, 1, 2, 3, etc)

As Y is always positive,
we have
X + Y = Y
or X = 0
or x - 2 = 0
or x = 2

Hence, Statement 2) is alone sufficient

Rgds,
Rajat
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##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 56244
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

This question is discussed here: new-set-number-properties-149775.html

If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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2
This is one of the classic question.
(1). |x−2|<2−y. The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus ,2-y>0 thus y<2 . Next, since given that is a positive integer, then y=1.

So, we have that:|x-2|<1 , which implies that ,-1<x-2>1 thus x=2. Sufficient.

(2).x+y-3 = |1-y|

we can write this in two form , considering positive & negative

(a) x+y-3 = 1-y
x=1-y-y+3 => x= 1-2y+3 => x= 2(2-y)
Since x is positive integer so y can be greater than 2 so y has to be 1. So x=2

(b) x+y-3 = y-1
x=2.
So by both ways X=2.
Each statement is sufficient
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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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Hi

Isn't 0 a positive interger. If isn't whenever the question mentions positive or negative intergers we do not take 0...right

what about non-negative integers which include...0.1..2......

VeritasPrepKarishma wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

Math Expert V
Joined: 02 Sep 2009
Posts: 56244
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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akshaybansal991 wrote:
Hi

Isn't 0 a positive interger. If isn't whenever the question mentions positive or negative intergers we do not take 0...right

what about non-negative integers which include...0.1..2......

VeritasPrepKarishma wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

0 is neither positive nor negative integer (the only one of this kind).

Check for more here: number-properties-tips-and-hints-174996.html
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Joined: 30 Apr 2017
Posts: 60
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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VeritasPrepKarishma wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

Hi VeritasPrepKarishma

statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer!
Math Expert V
Joined: 02 Sep 2009
Posts: 56244
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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1
soodia wrote:
VeritasPrepKarishma wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

Hi VeritasPrepKarishma

statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer!

Yes, 0 is neither positive nor negative but (2) IS sufficient. Please check the OA under the spoiler and read the whole discussion: https://gmatclub.com/forum/if-x-and-y-a ... l#p1367359
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Posts: 9433
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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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soodia wrote:
VeritasPrepKarishma wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

Hi VeritasPrepKarishma

statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer!

Where did I assume/say that X or Y is 0?
_________________
Karishma
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Manager  B
Joined: 30 Apr 2017
Posts: 60
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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VeritasPrepKarishma wrote:
soodia wrote:
VeritasPrepKarishma wrote:
(1) |x−2|<2−y.

(2) x+y−3=|1−y|

"If x and y are positive integers" implies x and y are 1/2/3/4... etc

(1) |x−2|<2−y
Minimum value of y is 1 which means maximum value of right hand side is 2-1 = 1. An absolute value cannot be negative so left hand side must be 0 only (to be less than 1 but be an integer)

|x−2| = 0 means x = 2 (prime number)
This statement alone is sufficient.

(2) x+y−3=|1−y|
x = |1−y| - y + 3

If y = 1, x = 0 - 1 + 3 = 2
If y is greater than 1, (1-y) will be negative so |1−y| = -(1 - y) = y - 1
So x = y - 1 - y + 3 = 2

So for all cases, x must be 2. This statement alone is sufficient.

Hi VeritasPrepKarishma

statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer!

Where did I assume/say that X or Y is 0?[/quote]

WOW!
such a shame!!!
bunuel responded me too! but I did'n' find my mistake
I'm really sorry Karishma
this type of mistake will ruin my exam totally!
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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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1
From statement 1 : |x−2|<2−y [u][/u]
HERE WE HAVE 2 CASES
CASE 1 :
=> X-2< 2-Y FOR X≥2
=> X+Y< 4 ONLY POSSIBLE WHEN x = 2 (PRIME)
CASE 2 :
OR 2-X < 2-Y FOR X<2
=> X-Y>0 (ONLY POSSIBLE WHEN X IS 1 AND Y IS 0 BUT AS PER THE QUESTION X AND Y MUST BE POSITIVE INTEGERS AND 0 IS NOT A ONE )
THUS FROM STATEMENT 1 WE HAVE X=2 (PRIME) SUFFICIENT
FROM STATEMENT 2
WE HAVE 2 CASES HERE
CASE 1 :
X+Y-3 = 1-Y FOR Y ≤1
=> X+2Y = 4 THIS GIVES ONLY POSSIBLE VALUE FOR X AS 2 (PRIME)
CASE 2 :
X+Y-3= Y-1 FOR Y>1
THIS GIVES AGAIN X=2 (PRIME)
HENCE BOTH THE CASES GIVE X= 2 WHICH IS PRIME .
HENCE STATEMENT 2 IS ALSO SUFFICIENT
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Re: If x and y are positive integers, is x a prime number?  [#permalink]

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Bunuel wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

This question is discussed here: http://gmatclub.com/forum/new-set-numbe ... 49775.html

If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel,

I have a question in statement 1 :

We know X is positive so we open the MOD for X>0

X-2 < 2-y

X < 4-y

X+y < 4

As we know 0 is niether +ve nor -ve, we are left out with values for X and y to be 1,2,3

Why do we have to only consider x=2 here?

Why can't the values be x=1, y=2 or x=1,y=1?
They also satisfy the equation we got from statement 1
Math Expert V
Joined: 02 Sep 2009
Posts: 56244
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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Pratyaksh2791 wrote:
Bunuel wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

This question is discussed here: http://gmatclub.com/forum/new-set-numbe ... 49775.html

If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel,

I have a question in statement 1 :

We know X is positive so we open the MOD for X>0

X-2 < 2-y

X < 4-y

X+y < 4

As we know 0 is niether +ve nor -ve, we are left out with values for X and y to be 1,2,3

Why do we have to only consider x=2 here?

Why can't the values be x=1, y=2 or x=1,y=1?
They also satisfy the equation we got from statement 1

1. We are told that x is positive not x - 2, so you certainly cannot say that |x - 2| = x - 2 only because x is positive. For example, if x were 1, then |x - 2| = -(x - 2)
2. Neither x = 1 and y = 2 nor x = 1 and y = 1 satisfy |x - 2| < 2 - y.

The question above is not that easy. You should be absolutely clear with fundamentals before attempting such questions:

10. Absolute Value

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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If x and y are positive integers, is x a prime number?  [#permalink]

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Bunuel wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

This question is discussed here: http://gmatclub.com/forum/new-set-numbe ... 49775.html

If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel,

I was unable to understand, this part how did you get "x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$
I understand the theory of absolute value, have gone through theory in Gmatclub as well Egmat articles, but Unable to get this part ?
Math Expert V
Joined: 02 Sep 2009
Posts: 56244
Re: If x and y are positive integers, is x a prime number?  [#permalink]

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hero_with_1000_faces wrote:
Bunuel wrote:
bekerman wrote:
If x and y are positive integers, is x a prime number? [M28-58]

(1) |x−2|<2−y.

(2) x+y−3=|1−y|

This question is discussed here: http://gmatclub.com/forum/new-set-numbe ... 49775.html

If x and y are positive integers, is x a prime number?

(1) |x - 2| < 2 - y . The left hand side of the inequality is an absolute value, so the least value of LHS is zero, thus 0 < 2 - y, thus y < 2 (if y is more than or equal to 2, then $$y-2\leq{0}$$ and it cannot be greater than |x - 2|). Next, since given that y is a positive integer, then y=1.

So, we have that: $$|x - 2| < 1$$, which implies that $$-1 < x-2 < 1$$, or $$1 < x < 3$$, thus $$x=2=prime$$. Sufficient.

(2) x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$, thus $$|1-y|=-(1-y)$$. So, we have that $$x + y - 3 = -(1-y)$$, which gives $$x=2=prime$$. Sufficient.

Hi Bunuel,

I was unable to understand, this part how did you get "x + y - 3 = |1-y|. Since y is a positive integer, then $$1-y\leq{0}$$
I understand the theory of absolute value, have gone through theory in Gmatclub as well Egmat articles, but Unable to get this part ?

Positive integers are: 1, 2, 3, 4, 5, .... Thus $$y \geq 1$$, which is the same as $$1-y\leq{0}$$.
_________________ Re: If x and y are positive integers, is x a prime number?   [#permalink] 28 Dec 2018, 00:13
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