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)