If (a + 1) and (a - 1) are both prime numbers and "a" is a

Question

If (a + 1) and (a - 1) are both prime numbers and "a" is a natural number, then a is? 

Necessarily an even number 
necessarily divisible by 3 
an even number and is divisible by 3 
no such number exists 
none of these

gpmagnus · Answer

Let's take a look at a list of primes upto 100 or so:

2  3 5 7 11 13  17 19 23  29 31  37  41 43  47 53  59 61  67  71 73  79 83 89 97  101 103 

I've put in boldface those primes which fit the (a-1), (a+1) criterion. Let's make a list to check for a:

3, 5 -> a=4 --> a is even but not necessarily divisible by 3 

 => answers B, C and D are out. this means we basically need to 
 see if a is always even. If so, answer will be A. If not - E.

We can check the other pairs of primes, but it is not neccessary. Other than the number 2, all other primes are odd, so we can express them as follows:

a-1 = prime = 2k+1 --> a = 2k+2 = 2(k+1) --> ergo a is even

a+1 = prime = 2k+1 --> a = 2k --> ergo a is even

Thus the answer is A

chuckle · Answer

I will go with "None of these".
if a = 8, a+1 is not prime. Hence it is definitely not A.

vivek123 · Answer

I guess the stem says, "a" is such that (a+1) & (a-1) are both prime numbers.

a=8 is invalid consideration since 
(a-1) = 7 is a prime but 
(a+1)=9 is not a prime.

Matador · Answer

Answer A. Necessarily an even number.

Just pick numbers. If (a+1)  and  (a-1) are both prime numbers, we can pick numbers to solve for  a  in one of them and then plug  a  into the other one to see if we get a prime number there too. If so, then we get valid number for  a . Once we have two valid numbers for  a , we can look at their properties.

Pick the prime number 3:

(a+1) = 5 ---> a = 4

plug in 4 into the next expression

(a-1) = (4-1) = 3, which is a prime. Therefore, a can be 4.

Finding another number for a... pick a new prime, e.g. 13:

(a+1) = 13 ---> a = 12

(a-1) = (12-1) = 11, which is a prime. Therefore, a can be 12 as well.

We now have two valid cases:
a = 4
a = 12

In both cases, answer A applies. Thus, the correct answer is A.

ywilfred · Answer

x and y are the prime numbers
a-1 = x --> a = x + 1
a+1 = y --> a = y -1

x+1 = y-1
y-x = 2

y-x = 2
5-3 = 2 --> a = 5-1 = 4, a = 3+1 = 4 Even.
7-5 = 2 --> a = 7-1 = 6, a = 5+1 = 6 Even.

A is the best choice.
(B) --> first option is not divisible by 3
(C) --> Same as B
(D) and (E) are out since we can arrive at a choice.

GMATT73 · Answer

(A) has to be true given the operations (a+1) and (a-1) both result in a prime number.

INSEAD1979 · Answer

All primes are odd except for 2 .

If a-1 and a+1 are prime and hence odd. A has to be even. That is it.(check 3-1 and 3+1 to see that 4 is not prime so a can not be 3).

Keep it simple.

If (a + 1) and (a - 1) are both prime numbers and "a" is a

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