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

Question

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

Bunuel · Accepted Answer

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.

atalpanditgmat · Answer

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

yezz · Answer

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)

Bunuel · Answer

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.

jlgdr · Answer

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

Blackishmamba · Answer

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

goldbill · Answer

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

thangvietnam · Answer

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

shyam30 · 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!

Posted from my mobile device

chetan2u · Answer

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

siddhantvarma · Answer

Given: x>0, x is an integer (x=1,2,3,....)
We need to check if x is a prime number

Statement 1: x-1 is a prime number
x = 3, 3-1 = 2 
x = 4, 4-1 = 3 
Thus x can be either 3 (prime) or 4 (not prime) => Statement 1 is not sufficient

Statement 2: x-5 is a prime number
x = 7, 7 - 5 = 2
x = 8, 8-5 = 3
Thus x can be either 7 (prime) or 8 (not prime) => Statement 2 is not sufficient

Combing the two:
We have x - 1 = A and x - 5 = B where A and B are prime numbers (it's clear by looking at these equations that A ≠ B)

Subtract the two equations;
(x-1) - (x-5) = A - B => A - B = 4 
Add the two equations;
2x - 6 = A + B => x = (A + B + 6)/2

Neither A not B can be 2 here as:
If A = 2, x - 1 = 2 => x = 3, but 3-5 = -2 has to be a prime which it isn't
Similarly, if B = 2, x - 5 = -3, not a positive integer
Means A and B are odd prime numbers whose difference is 4

A and B can be: 11 and 7, 17 and 13, 23 and 19, ....
x can be 11+7+6/2 = 12, 17+13+6/2 = 18, and so on

Clearly, x will always be an even number => x will never be prime
=> Therefore both statements together are sufficient

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

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