If three prime numbers, all greater than 
, are in A.P. , then their common difference
(A) must be divisible by 
but not necessarily by ;
(B) must be divisible by but not necessarily by ;
(C) must be divisible by both and ;
(D) need not be divisible by any of and ;
Discussion:
Say 
and 
are the three primes in A.P. with d as the common difference.
Since 
, hence 
is odd.
If 
is odd, then 
is even. But then cannot be a prime any more. Hence contradiction. Thus cannot be odd. Hence divides .
Again 
implies is not divisible by . Hence p is either 
or modulo . Now we wish to check if d is divisible by or not. Suppose it is not then d is also either or modulo .
Case 1: is mod and is mod , then is 
mod hence contradiction (as p+2d is a prime greater than 3).
Case 2: is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3)
Case 1: is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3).
Case 2: is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3)
Hence must be modulo .
Therefore common difference must be divisible by both and .
Answer: (C)
If three prime numbers, all greater than , are in A.P. , then their common difference
(A) must be divisible by but not necessarily by ;
(B) must be divisible by but not necessarily by ;
(C) must be divisible by both and ;
(D) need not be divisible by any of and ;
Discussion:
Say and are the three primes in A.P. with d as the common difference.
Since , hence is odd.
If is odd, then is even. But then cannot be a prime any more. Hence contradiction. Thus cannot be odd. Hence divides .
Again implies is not divisible by . Hence p is either or modulo . Now we wish to check if d is divisible by or not. Suppose it is not then d is also either or modulo .
Case 1: is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3).
Case 2: is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3)
Case 1: is mod and is mod , then is mod hence contradiction (as p+d is a prime greater than 3).
Case 2: is mod and is mod , then is mod hence contradiction (as p+2d is a prime greater than 3)
Hence must be modulo .
Therefore common difference must be divisible by both and .
Answer: (C)
