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

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.

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

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......

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

Check for more here: number-properties-tips-and-hints-174996.html

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

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

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!

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

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

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.

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 \(1-y\leq{0}\).

I have done this problem following way-
Q: If x and y are positive integers, is x a prime number?
(1) |x−2|<2−y
Because x-2 can be positive or negative, let's first assume: it's positive,
So, x-2<2-y ---> x-4<-y ---> x+y<4.......(1)
Now let's assume: it's negative,
So, -x+2<2-y ---> y-x<2-2 ----> y<x. .....(2)
If we combine 1&2 , we get, x=2 (prime) & y=1, so, Stmt 1 is sufficient.

(2) x+y−3=|1−y|
Because 1-y can be positive or negative, let's first assume: it's positive,
So,x+y−3=1−y---> x+y-3-1+y= 0----> x+2y-4=0 ---> x+2y=4.......(3)
Now let's assume: it's negative,
So,x+y−3=-1+y ---> x+y-3+1-y= 0 -----> x-2=0..So, x=2.........(4)
If we combine 3&4 , again we get, x=2 (prime) , y=1; so, Stmt 2 is also sufficient.
So, choice D is correct: EACH statement ALONE is sufficient.

Hi All,

Many DS questions can be solved by TESTing VALUES - and this prompt gives us a great opportunity to do so. As with any question, you have to pay attention to the 'restrictions' that the prompt places on us, including the fact that the result of an Absolute Value calculation will NEVER be a negative number (that result will either be a positive number or a 0).

If X and Y are POSITIVE INTEGERS, is X a PRIME number?

1) |X−2| < 2−Y

2) X+Y−3 = |1−Y|

Here, we're restricted to positive integers ONLY and we're asked if X is a PRIME number. This is a YES/NO question.

1) |X−2| < 2−Y

Notice that the 'left side' of this inequality includes an Absolute Value, so that part of the calculation cannot be any smaller than 0. This is interesting, since the 'right side' must be GREATER than the 'left side', but we are SUBTRACTING a positive number from 2. This severely limits the value of Y.

Y CANNOT be 3.... since 2-3 = -1... and that won't be greater than the 'left side'
Y CANNOT be 2.... since 2-2 = 0... and that won't be greater than the 'left side'

Remember that Y MUST be a positive integer, so there's only one option left. Y MUST be 1. We can 'lock' that value in place.

Y = 1

Now we have...
|X-2| < 1

Again, we're restricted to positive integers only for X - and there's only one possible value for X that will fit this inequality: it's when X = 2. This would give us...

|2-2| < 1
0 < 1

Since X = 2, the answer to the question is YES... and since there's only one possible answer, the answer is really ALWAYS YES.
Fact 1 is SUFFICIENT.

Notice how no special calculations were needed here; just a bit of Arithmetic and TESTing VALUES to determine what options are possible. Fact 2 will require a little more work, but the type of work will be the same.

2) X+Y−3 = |1−Y|

Here, the 'right side' of the equation can never be any smaller than 0, but the 'left side' includes a "-3", so we have to be sure that the result of that part of the calculation never drops below 0. Let's try TESTing some values and see what happens...Remember that both X and Y must be POSITIVE INTEGERS.

IF.... Y = 1
X+1-3 = |1-1|
X - 2 = 0
X = 2... Here, the answer to the question is YES.

IF.... Y = 2
X+2-3 = |2-1|
X - 1 = 1
X = 2... Here, the answer to the question is YES.

IF.... Y = 3
X+3-3 = |3-1|
X = 2
X = 2... Here, the answer to the question is YES.

Notice the pattern? If you're not sure yet, then feel free to try a few more TESTs (since you're just doing basic Arithmetic, that additional work shouldn't take too long. The answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer:

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