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# If x and z are positive integers, is at least one of them a prime numb

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Math Expert
Joined: 02 Sep 2009
Posts: 58453
If x and z are positive integers, is at least one of them a prime numb  [#permalink]

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01 Oct 2018, 04:53
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Difficulty:

95% (hard)

Question Stats:

39% (02:26) correct 61% (02:12) wrong based on 77 sessions

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If x and z are positive integers, is at least one of them a prime number?

(1) x^2 = 15 + z^2
(2) (x − z) is a prime number

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Re: If x and z are positive integers, is at least one of them a prime numb  [#permalink]

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01 Oct 2018, 05:17
Bunuel wrote:
If x and z are positive integers, is at least one of them a prime number?

(1) x^2 = 15 + z^2
(2) (x − z) is a prime number

Question: is at least one of x and z a prime number?

Statement 1:x^2 = 15 + z^2
i.e. $$x^2-z^2 =15$$
i.e. $$(x-z)*(x+z) =15$$
i.e. $$(x-z)*(x+z) =3*5$$ OR
i.e. $$(x-z)*(x+z) =1*15$$
If x-z = 3 the x and z may be 5 and 2 both prime OR x and z may be 9 and 6 both NON Prime
NOT SUFFICIENT

Statement 2: (x − z) is a prime number
x-z = 5-3=2 i.e. Both prime OR
x-z = 6-4=2 i.e. None prime
NOT SUFFICIENT

Combining both statements
$$(x-z)*(x+z) =15$$ and $$x-z$$ is prime i.e. either 3 or 5
If x-z = 3 the x and z may be 5 and 2 both prime OR x and z may be 9 and 6 both NON Prime
NOT SUFFICIENT

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Re: If x and z are positive integers, is at least one of them a prime numb  [#permalink]

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01 Oct 2018, 21:27
Bunuel wrote:
If x and z are positive integers, is at least one of them a prime number?

(1) x^2 = 15 + z^2
(2) (x − z) is a prime number

Lets deal with Statement 1 first using plugin approach:

1 x^2 = 15 +z^2

2 we can write above statement as

x^2-Z^2=15 or (x+z)(x-z)=15

3. x+z= 3 means- 1 + 2=3 and x-z means- 4-1

4. so (1,2) and (4,1) -leads to insufficent we have 2 answers

A insufficient

II. Lets move on to 2nd statement

1. x-z is prime number
2. put x=4 , z=1 , none are prime , 4-1=3 , which is prime, so we got 1 no answer
3. put x=5 , y=2, both are prime, 5-2=3, which is prime, so we got 1 yes answer

So clearly B is insufficent

III. combining Statement 1 and statement 2

a)x^2-z^2=15
b) x-z =prime

if you plug values (4,1), both satisfies statement a and b,
if you try value like(5,2) , or any other combination, it wont satisfy both

hence A and B combined are sufficnet

Bunuel, please correct my approach if i m wrong
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Re: If x and z are positive integers, is at least one of them a prime numb  [#permalink]

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01 Oct 2018, 23:28
Given that x and z are positive integers.
We need to find out is that whether one of those number or numbers is a prime number.
Statement 1 says that
x^2 = 15+ z^2
This could happen in 2 of the most nearest cases, nearest in the sense smaller numbers.
if x = 4 , then x^2 = 16, then
16 = 15 + z^2
Implies, z^2 = 1,
Implies, z= 1(since z is a positive integer)
So , x = 4, z = 1, neither of them are prime numbers.
Case 2 , if x = 8, then x^2 = 64,
Implies, 64 = 15 + z^2
Implies, z^2 = 49
Implies, z =7 (since z is a positive integer)
So, x = 8, z = 7 (one of them is a prime number)
Since we have no unique solution of one of the or both of them to be prime numbers. Statement 1 is insufficient.
Statement 2 says that x-z is a prime number.
Now, if both x and z are odd, then the only even prime number would be 2; and x should be equal to z+2, ie, x=z+2
Or if one of them is even then x-z will yield any odd prime number.
Per say this statement falls too short in providing data to find the solution, because, x-z could be 3-1(one of them prime), 7-5 (both of them prime),
So, statement 2 is insufficient.
Combining both statements we get that neither of them are supposed to be prime to arrive at the solution.
So the correct answer is C.
Re: If x and z are positive integers, is at least one of them a prime numb   [#permalink] 01 Oct 2018, 23:28
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