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

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

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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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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|



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

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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
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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.

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

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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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New post 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)

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

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New post 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
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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

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New post 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!

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

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New post 09 Nov 2017, 21:27
1
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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
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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

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

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New post 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.

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?[/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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New post 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 )
HENCE DISCARD CASE 2
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
HENCE ANSWER IS D

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