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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? [M2858] (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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bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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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22 May 2014, 23:00
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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, 2y > 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 = 2Hence, Statement 1) is alone sufficient2) x+y−3=1−y Can be written as: x+y3 =y−1 (Same as it is modulus) or x  2 + (y 1) = y−1 Using, X + Y = Y Where x2 = 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 = 2Hence, Statement 2) is alone sufficientAnswer (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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23 May 2014, 01:53
bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(1) x−2<2−y.
(2) x+y−3=1−y This question is discussed here: newsetnumberproperties149775.htmlIf 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 \(y2\leq{0}\) 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 \(1 < x2 < 1\), or \(1 < x < 3\), thus \(x=2=prime\). Sufficient. (2) x + y  3 = 1y. Since y is a positive integer, then \(1y\leq{0}\), thus \(1y=(1y)\). So, we have that \(x + y  3 = (1y)\), 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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05 Jul 2014, 07:23
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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 ,2y>0 thus y<2 . Next, since given that is a positive integer, then y=1. So, we have that:x2<1 , which implies that ,1<x2>1 thus x=2. Sufficient. (2).x+y3 = 1y we can write this in two form , considering positive & negative (a) x+y3 = 1y x=1yy+3 => x= 12y+3 => x= 2(2y) Since x is positive integer so y can be greater than 2 so y has to be 1. So x=2 (b) x+y3 = y1 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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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 nonnegative integers which include...0.1..2...... VeritasPrepKarishma wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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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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 nonnegative integers which include...0.1..2...... VeritasPrepKarishma wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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: numberpropertiestipsandhints174996.html
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Re: If x and y are positive integers, is x a prime number? [#permalink]
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09 Nov 2017, 18:27
VeritasPrepKarishma wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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 VeritasPrepKarishmastatement 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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09 Nov 2017, 21:27
soodia wrote: VeritasPrepKarishma wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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 VeritasPrepKarishmastatement 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/ifxandya ... l#p1367359
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Re: If x and y are positive integers, is x a prime number? [#permalink]
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09 Nov 2017, 22:37
soodia wrote: VeritasPrepKarishma wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(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 21 = 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, (1y) 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 VeritasPrepKarishmastatement 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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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 21 = 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, (1y) 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 VeritasPrepKarishmastatement 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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10 Nov 2017, 12:21
From statement 1 : x−2<2−y [u][/u] HERE WE HAVE 2 CASES CASE 1 : => X2< 2Y FOR X≥2 => X+Y< 4 ONLY POSSIBLE WHEN x = 2 (PRIME) CASE 2 : OR 2X < 2Y FOR X<2 => XY>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+Y3 = 1Y FOR Y ≤1 => X+2Y = 4 THIS GIVES ONLY POSSIBLE VALUE FOR X AS 2 (PRIME) CASE 2 : X+Y3= Y1 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]
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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? [M2858]
(1) x−2<2−y.
(2) x+y−3=1−y This question is discussed here: http://gmatclub.com/forum/newsetnumbe ... 49775.htmlIf 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 \(y2\leq{0}\) 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 \(1 < x2 < 1\), or \(1 < x < 3\), thus \(x=2=prime\). Sufficient. (2) x + y  3 = 1y. Since y is a positive integer, then \(1y\leq{0}\), thus \(1y=(1y)\). So, we have that \(x + y  3 = (1y)\), which gives \(x=2=prime\). Sufficient. Answer: D. Hi Bunuel, I have a question in statement 1 : We know X is positive so we open the MOD for X>0 X2 < 2y X < 4y 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



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Re: If x and y are positive integers, is x a prime number? [#permalink]
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11 Dec 2017, 05:13
Pratyaksh2791 wrote: Bunuel wrote: bekerman wrote: If x and y are positive integers, is x a prime number? [M2858]
(1) x−2<2−y.
(2) x+y−3=1−y This question is discussed here: http://gmatclub.com/forum/newsetnumbe ... 49775.htmlIf 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 \(y2\leq{0}\) 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 \(1 < x2 < 1\), or \(1 < x < 3\), thus \(x=2=prime\). Sufficient. (2) x + y  3 = 1y. Since y is a positive integer, then \(1y\leq{0}\), thus \(1y=(1y)\). So, we have that \(x + y  3 = (1y)\), which gives \(x=2=prime\). Sufficient. Answer: D. Hi Bunuel, I have a question in statement 1 : We know X is positive so we open the MOD for X>0 X2 < 2y X < 4y 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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