If x is a positive integer, is x a prime integer?

If x is a positive integer, is x a prime integer?

(1) x+1 is a prime number. Notice that x cannot be an odd prime, because in this case (odd prime) + 1 = (even number), and the only even prime is 2, but in this case x=1, which is not prime. So, if x=prime, then it must be 2: 2+1=3=prime. Of course, x can be a non-prime, for example 1. Not sufficient.

(2) x-5 is a prime number. If x=7, then the answer is YES but if x=8, then the answer is NO. Not sufficient.

(1)+(2) From (1) we have that if x IS a prime, then it must be 2. But if x=2, them x-5=-3, which is not a prime. Thus x=prime=2 contradicts the second statement. Therefore x cannot be a prime number. Sufficient.

Answer: C.

Hope it's clear.
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Hi Bunuel, My approach does not look standard. Is it correct?

From statement 1: X= 1,2,4,6,10,12,16,18........
From statement 2: X= 7,8,10,12,16,22........

common from the both statements are 10,12,16..... Thus X is not a prime integer.

Kindly provide algebric approach if possible.

Thanks in advance

f x is a positive integer, is x a prime integer?

(1) x+1 is a prime number.
(2) x-5 is a prime number.


I think this is a bad question , however ...

from 1

x , x+1 are 2 successive integers ....... if x is prime and since the only 2 prime that are successive are 2,3 therefore if x is prime it is 2 , however x could be 4 as well .... insuff

from 2

The only 2 primes 5 units apart are 7 and 2 , (since all primes > 3 end in 1,3,7,9) if x were to be prime it is 7 , however x could be 8 and (x-5) could be 3 ... insuff

both together

for x to be prime according to the above 2 statements then it has be less than a prime by 1 and larger than another by 5 .... there exist no such a prime

thus sure x is not a prime ( integers if not prime are composite)

Yes, that's correct. From both statements x must be even (10, 12, 16, ...), but it cannot be 2. So, x is not a prime number.
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Correct. From both statements together:

X must be higher than 5 because no prime can be negative. All primes higher than 5 are odd. So x+1 will be an even number. There is only one even number that is prime which is 2. Given that x>5 then we know that X cannot be a prime

C

If x is a positive integer, is x a prime integer?

(1) x+1 is a prime number.
x+1 can be either of 2,3,5,7,11,13,17,19,... etc
Hence, X can be either of 1,2,4,6,10,12,16,18,...etc
Here X=2 is prime, while other values are non-prime
NOT SUFFICIENT

(2) x-5 is a prime number.
x-5 can be either of 2,3,5,7,11,13,17,19,... etc
Hence, X can be either of 7,8,10,12,16,18,22,24,...etc
Here X=7 is prime, while other values are non-prime
NOT SUFFICIENT

Considering (1) and (2) together,
X can be 10,12,16,18...etc
all of this values are even values greater than 2
hence, X cannot be prime
SUFFICIENT

Statement 1
x+1 is prime
when x =2, both x+1 and x are prime numbers
However when x= 4,6,10…, while x+1 is prime, x is not prime

Therefore, Statement 1 is insufficient

Statement 2
x-5 is prime
when x= 7 both x-5 and x are prime numbers
However when x= 8,10,12…, while x-5 is prime, x is not

Therefore Statement 2 is insufficient

Statement 1+2
Now for x to be prime it has to be equal to 2,3,5,7,11…
coming back to the question: is x a prime integer?
if both x+1 and x-5 are prime numbers:
then x cannot be equal to 2, the only prime number which makes x+1 a prime
also x cannot be equal to 7, the only prime number which makes x-5 a prime

for both statements to be true x has to be a non prime value

Hence 1+2 is sufficient

consider both statements.
look at condition 2.
x-5=2 is the only prime which is even. x=7
x-5= other prime>2, this prime must be odd, and x must be even.

x=7 can not satisfy condition 1. so, x must be even such as x=10. so, x can not be a prime. choice C is answer

I think this question is wrong!

Let's test two cases for potential values of x:

Case 1 (x=2 a prime number):
(1) x+1 is a prime number. -> 2+1 = 3 also a prime
(2) x-5 is a prime number. -> 2-5 = -3 also a prime

Case 2 (x=10 a composite number):
(1) x+1 is a prime number. -> 10+1 = 11 also a prime
(2) x-5 is a prime number. -> 10-5 = 5 also a prime

Since both x=2 and x=10 are valid cases, option E is correct!

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If x is a positive integer, is x a prime integer?

x is positive and an integer.

(1) x + 1 is a prime number.
When can x and x+1 be prime….ONLY when x is 2 as only consecutive numbers that are prime are 2 and 3.
If x is 2, answer is yes otherwise no.
Insufficient

(2) x - 5 is a prime number.
When can x and x-5 be prime…. Now, one of x and x-5 will be even and other odd.
ONLY even prime is 2, so the pair will be 2 and 7, where x is 7.
If x is 7, answer is yes otherwise no.
Insufficient


Combined
x is neither 2 nor 7. Hence, answer is NO always.


C
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