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

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If x and y are positive integers, is x a prime number? 22 May 2014, 21:48
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bekerman
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|
Show more


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

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If x and y are positive integers, is x a prime number? 23 May 2014, 01:53
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bekerman
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|
Show more

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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If x and y are positive integers, is x a prime number? 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 ,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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If x and y are positive integers, is x a prime number? 21 Aug 2014, 23:23
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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
bekerman
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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If x and y are positive integers, is x a prime number? 22 Aug 2014, 04:01
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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
bekerman
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)
Show more

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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If x and y are positive integers, is x a prime number? 09 Nov 2017, 18:27
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bekerman
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)
Show more


Hi VeritasPrepKarishma

statement 2 cannot be sufficient, question says X AND Y are positive, 0 is not the positive integer!
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If x and y are positive integers, is x a prime number? 09 Nov 2017, 21:27
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VeritasPrepKarishma
bekerman
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!
Show more

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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If x and y are positive integers, is x a prime number? 09 Nov 2017, 22:37
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bekerman
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!
Show more

Where did I assume/say that X or Y is 0?
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If x and y are positive integers, is x a prime number? 09 Nov 2017, 23:58
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soodia
VeritasPrepKarishma

(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!
Show more

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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If x and y are positive integers, is x a prime number? 10 Nov 2017, 12:21
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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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If x and y are positive integers, is x a prime number? 11 Dec 2017, 04:54
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Bunuel
bekerman
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: https://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.

Answer: D.
Show more
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
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If x and y are positive integers, is x a prime number? 11 Dec 2017, 05:13
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Bunuel
bekerman
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: https://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.

Answer: 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
Show more

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? 27 Dec 2018, 21:21
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Bunuel
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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|

This question is discussed here: https://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.

Answer: D.
Show more


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 ?
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If x and y are positive integers, is x a prime number? 28 Dec 2018, 00:13
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Bunuel
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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|

This question is discussed here: https://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.

Answer: D.


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 ?
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Positive integers are: 1, 2, 3, 4, 5, .... Thus \(y \geq 1\), which is the same as \(1-y\leq{0}\).
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If x and y are positive integers, is x a prime number? 25 Sep 2020, 20:04
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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.
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If x and y are positive integers, is x a prime number? 10 Sep 2021, 13:40
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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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D

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If x and y are positive integers, is x a prime number? 30 Jul 2025, 09:14
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