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

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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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22 May 2014, 17:42
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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|
[Reveal] Spoiler: OA

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Please +1 KUDOS if my post helps. Thank you.

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8028
Location: Pune, India
Re: If x and y are positive integers, is x a prime number? [#permalink]

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22 May 2014, 21:48
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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|

"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.

_________________

Karishma
Veritas Prep | GMAT Instructor
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 20 May 2014 Posts: 37 Location: India Schools: IIMC GMAT 1: 700 Q51 V32 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 22 May 2014, 23:00 3 This post received KUDOS 1 This post was BOOKMARKED 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 Answer (D) Rgds, Rajat _________________ If you liked the post, please press the'Kudos' button on the left Math Expert Joined: 02 Sep 2009 Posts: 44588 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 23 May 2014, 01:53 Expert's post 1 This post was BOOKMARKED 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. Answer: D. _________________ Manager Joined: 20 Jan 2014 Posts: 165 Location: India Concentration: Technology, Marketing Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 05 Jul 2014, 07:23 2 This post received KUDOS 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 _________________ Consider +1 Kudos Please Intern Joined: 05 May 2014 Posts: 4 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 21 Aug 2014, 23:23 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. Answer (D) Math Expert Joined: 02 Sep 2009 Posts: 44588 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 22 Aug 2014, 04:01 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. Answer (D) 0 is neither positive nor negative integer (the only one of this kind). Check for more here: number-properties-tips-and-hints-174996.html _________________ Manager Joined: 30 Apr 2017 Posts: 75 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 09 Nov 2017, 18:27 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. Answer (D) Hi VeritasPrepKarishma statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer! Math Expert Joined: 02 Sep 2009 Posts: 44588 Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 09 Nov 2017, 21:27 1 This post received KUDOS Expert's post 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. Answer (D) 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 _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8028 Location: Pune, India Re: If x and y are positive integers, is x a prime number? [#permalink] ### Show Tags 09 Nov 2017, 22:37 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. Answer (D) 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 Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Manager
Joined: 30 Apr 2017
Posts: 75
Re: If x and y are positive integers, is x a prime number? [#permalink]

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09 Nov 2017, 23:58
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!
Manager
Joined: 20 Feb 2017
Posts: 71
Re: If x and y are positive integers, is x a prime number? [#permalink]

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10 Nov 2017, 12:21
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
Intern
Joined: 06 Oct 2017
Posts: 10
Re: If x and y are positive integers, is x a prime number? [#permalink]

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11 Dec 2017, 04:54
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
Joined: 02 Sep 2009
Posts: 44588
Re: If x and y are positive integers, is x a prime number? [#permalink]

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11 Dec 2017, 05:13
Expert's post
1
This post was
BOOKMARKED
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.
_________________
Re: If x and y are positive integers, is x a prime number?   [#permalink] 11 Dec 2017, 05:13
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